<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a triangle \(ABC\) and an arbitrary point \(P\), the feet of the perpendiculars dropped from \(P\) to the sides of triangle \(ABC\) become the vertices of triangle \(A_1B_1C_1\). The vertices of triangle \(A_2B_2C_2\) are the intersections of lines \(AP\), \(BP\), and \(CP\) with the circumcircle of triangle \(ABC\), other than the points \(A\), \(B\), and \(C\). Prove that triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar. How many such points \(P\) exist for a scalene triangle \(ABC\) such that the corresponding triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar to triangle \(ABC\)?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a triangle \(ABC\) and an arbitrary point \(P\), the feet of the perpendiculars dropped from \(P\) to the sides of triangle \(ABC\) become the vertices of triangle \(A_1B_1C_1\). The vertices of triangle \(A_2B_2C_2\) are the intersections of lines \(AP\), \(BP\), and \(CP\) with the circumcircle of triangle \(ABC\), other than the points \(A\), \(B\), and \(C\). Prove that triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar. How many such points \(P\) exist for a scalene triangle \(ABC\) such that the corresponding triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar to triangle \(ABC\)?<|im_end|>
<|im_start|>assistant
<Z_369><Z_232><Z_294><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a triangle \(ABC\) and an arbitrary point \(P\), the feet of the perpendiculars dropped from \(P\) to the sides of triangle \(ABC\) become the vertices of triangle \(A_1B_1C_1\). The vertices of triangle \(A_2B_2C_2\) are the intersections of lines \(AP\), \(BP\), and \(CP\) with the circumcircle of triangle \(ABC\), other than the points \(A\), \(B\), and \(C\). Prove that triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar. How many such points \(P\) exist for a scalene triangle \(ABC\) such that the corresponding triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar to triangle \(ABC\)?<|im_end|>
<|im_start|>assistant
<Z_131><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a triangle \(ABC\) and an arbitrary point \(P\), the feet of the perpendiculars dropped from \(P\) to the sides of triangle \(ABC\) become the vertices of triangle \(A_1B_1C_1\). The vertices of triangle \(A_2B_2C_2\) are the intersections of lines \(AP\), \(BP\), and \(CP\) with the circumcircle of triangle \(ABC\), other than the points \(A\), \(B\), and \(C\). Prove that triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar. How many such points \(P\) exist for a scalene triangle \(ABC\) such that the corresponding triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar to triangle \(ABC\)?<|im_end|>
<|im_start|>assistant
<Z_96><Z_334><Z_409><ANSWER>

Question: Given a triangle \(ABC\) and an arbitrary point \(P\), the feet of the perpendiculars dropped from \(P\) to the sides of triangle \(ABC\) become the vertices of triangle \(A_1B_1C_1\). The vertices of triangle \(A_2B_2C_2\) are the intersections of lines \(AP\), \(BP\), and \(CP\) with the circumcircle of triangle \(ABC\), other than the points \(A\), \(B\), and \(C\). Prove that triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar. How many such points \(P\) exist for a scalene triangle \(ABC\) such that the corresponding triangles \(A_1B_1C_1\) and \(A_2B_2C_2\) are similar to triangle \(ABC\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miriam currently has 30 marbles, but she used to have more. Miriam gave her brother 60 marbles, gave her sister twice that amount and gave her friend Savanna three times the amount she currently has. How many marbles did Miriam start with?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miriam currently has 30 marbles, but she used to have more. Miriam gave her brother 60 marbles, gave her sister twice that amount and gave her friend Savanna three times the amount she currently has. How many marbles did Miriam start with?<|im_end|>
<|im_start|>assistant
<Z_369><Z_89><Z_253><Z_200><Z_244><Z_470><Z_368><Z_445><Z_279><Z_433><Z_278><Z_329><Z_138><Z_232><Z_418><Z_378><Z_440><Z_299><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miriam currently has 30 marbles, but she used to have more. Miriam gave her brother 60 marbles, gave her sister twice that amount and gave her friend Savanna three times the amount she currently has. How many marbles did Miriam start with?<|im_end|>
<|im_start|>assistant
<Z_483><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Miriam currently has 30 marbles, but she used to have more. Miriam gave her brother 60 marbles, gave her sister twice that amount and gave her friend Savanna three times the amount she currently has. How many marbles did Miriam start with?<|im_end|>
<|im_start|>assistant
<Z_31><Z_129><Z_136><Z_101><Z_165><Z_421><Z_478><Z_312><Z_135><Z_358><Z_83><Z_292><Z_104><Z_277><Z_449><Z_197><Z_315><Z_257><ANSWER>

Question: Miriam currently has 30 marbles, but she used to have more. Miriam gave her brother 60 marbles, gave her sister twice that amount and gave her friend Savanna three times the amount she currently has. How many marbles did Miriam start with?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Allyn is a golfer. At the starting tee, he hit the golf ball and it traveled some distance straight toward the hole. On his second turn, he hit the ball again straight toward the hole and it traveled half as far as it did on his first turn, but the ball landed 20 yards beyond the hole. The distance from the starting tee to the hole is 250 yards. How far did the ball travel on Allyn's first turn?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Allyn is a golfer. At the starting tee, he hit the golf ball and it traveled some distance straight toward the hole. On his second turn, he hit the ball again straight toward the hole and it traveled half as far as it did on his first turn, but the ball landed 20 yards beyond the hole. The distance from the starting tee to the hole is 250 yards. How far did the ball travel on Allyn's first turn?<|im_end|>
<|im_start|>assistant
<Z_477><Z_151><Z_267><Z_70><Z_240><Z_322><Z_168><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Allyn is a golfer. At the starting tee, he hit the golf ball and it traveled some distance straight toward the hole. On his second turn, he hit the ball again straight toward the hole and it traveled half as far as it did on his first turn, but the ball landed 20 yards beyond the hole. The distance from the starting tee to the hole is 250 yards. How far did the ball travel on Allyn's first turn?<|im_end|>
<|im_start|>assistant
<Z_298><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Allyn is a golfer. At the starting tee, he hit the golf ball and it traveled some distance straight toward the hole. On his second turn, he hit the ball again straight toward the hole and it traveled half as far as it did on his first turn, but the ball landed 20 yards beyond the hole. The distance from the starting tee to the hole is 250 yards. How far did the ball travel on Allyn's first turn?<|im_end|>
<|im_start|>assistant
<Z_475><Z_0><Z_35><Z_231><Z_446><Z_506><Z_391><Z_150><ANSWER>

Question: Allyn is a golfer. At the starting tee, he hit the golf ball and it traveled some distance straight toward the hole. On his second turn, he hit the ball again straight toward the hole and it traveled half as far as it did on his first turn, but the ball landed 20 yards beyond the hole. The distance from the starting tee to the hole is 250 yards. How far did the ball travel on Allyn's first turn?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 3x^2 + 12x - 15$.

(1) Find the zeros of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ on the interval \[-3, 3\]; (3) Prove that $f(x)$ is an increasing function on the interval \[-2, +\infty\).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 3x^2 + 12x - 15$.

(1) Find the zeros of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ on the interval \[-3, 3\]; (3) Prove that $f(x)$ is an increasing function on the interval \[-2, +\infty\).<|im_end|>
<|im_start|>assistant
<Z_24><Z_449><Z_322><Z_461><Z_97><Z_263><Z_430><Z_427><Z_250><Z_504><Z_278><Z_139><Z_213><Z_151><Z_188><Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 3x^2 + 12x - 15$.

(1) Find the zeros of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ on the interval \[-3, 3\]; (3) Prove that $f(x)$ is an increasing function on the interval \[-2, +\infty\).<|im_end|>
<|im_start|>assistant
<Z_203><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given the function $f(x) = 3x^2 + 12x - 15$.

(1) Find the zeros of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ on the interval \[-3, 3\]; (3) Prove that $f(x)$ is an increasing function on the interval \[-2, +\infty\).<|im_end|>
<|im_start|>assistant
<Z_381><Z_240><Z_226><Z_507><Z_208><Z_196><Z_238><Z_13><Z_357><Z_499><Z_264><Z_176><Z_453><Z_153><Z_435><Z_52><ANSWER>

Question:
Given the function $f(x) = 3x^2 + 12x - 15$.

(1) Find the zeros of $f(x)$; (2) Find the maximum and minimum values of $f(x)$ on the interval \[-3, 3\]; (3) Prove that $f(x)$ is an increasing function on the interval \[-2, +\infty\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Monica was studying for an exam. She studied for 2 hours on Wednesday, three times as long on Thursday, and half of the time she studied on Thursday on Friday. During the weekend (two days) Monica studied for some hours. She spent a total of 22 hours studying during the five days. What is the ratio of the time she studied during the weekend to the time she studied on Wednesday, Thursday, and Friday combined?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Monica was studying for an exam. She studied for 2 hours on Wednesday, three times as long on Thursday, and half of the time she studied on Thursday on Friday. During the weekend (two days) Monica studied for some hours. She spent a total of 22 hours studying during the five days. What is the ratio of the time she studied during the weekend to the time she studied on Wednesday, Thursday, and Friday combined?<|im_end|>
<|im_start|>assistant
<Z_477><Z_211><Z_236><Z_293><Z_185><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Monica was studying for an exam. She studied for 2 hours on Wednesday, three times as long on Thursday, and half of the time she studied on Thursday on Friday. During the weekend (two days) Monica studied for some hours. She spent a total of 22 hours studying during the five days. What is the ratio of the time she studied during the weekend to the time she studied on Wednesday, Thursday, and Friday combined?<|im_end|>
<|im_start|>assistant
<Z_241><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Monica was studying for an exam. She studied for 2 hours on Wednesday, three times as long on Thursday, and half of the time she studied on Thursday on Friday. During the weekend (two days) Monica studied for some hours. She spent a total of 22 hours studying during the five days. What is the ratio of the time she studied during the weekend to the time she studied on Wednesday, Thursday, and Friday combined?<|im_end|>
<|im_start|>assistant
<Z_182><Z_464><Z_255><Z_35><Z_348><ANSWER>

Question: Monica was studying for an exam. She studied for 2 hours on Wednesday, three times as long on Thursday, and half of the time she studied on Thursday on Friday. During the weekend (two days) Monica studied for some hours. She spent a total of 22 hours studying during the five days. What is the ratio of the time she studied during the weekend to the time she studied on Wednesday, Thursday, and Friday combined?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?<|im_end|>
<|im_start|>assistant
<Z_369><Z_286><Z_67><Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?<|im_end|>
<|im_start|>assistant
<Z_46><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?<|im_end|>
<|im_start|>assistant
<Z_440><Z_393><Z_332><Z_491><ANSWER>

Question: What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To calculate $41^2$, John mentally computes the value of $40^2$ and adds a certain number. To compute $39^2$, he subtracts a number from $40^2$. What number does he subtract?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To calculate $41^2$, John mentally computes the value of $40^2$ and adds a certain number. To compute $39^2$, he subtracts a number from $40^2$. What number does he subtract?<|im_end|>
<|im_start|>assistant
<Z_24><Z_181><Z_301><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To calculate $41^2$, John mentally computes the value of $40^2$ and adds a certain number. To compute $39^2$, he subtracts a number from $40^2$. What number does he subtract?<|im_end|>
<|im_start|>assistant
<Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To calculate $41^2$, John mentally computes the value of $40^2$ and adds a certain number. To compute $39^2$, he subtracts a number from $40^2$. What number does he subtract?<|im_end|>
<|im_start|>assistant
<Z_274><Z_52><Z_412><ANSWER>

Question: To calculate $41^2$, John mentally computes the value of $40^2$ and adds a certain number. To compute $39^2$, he subtracts a number from $40^2$. What number does he subtract?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the l.c.m of 7/10, 8/9, 3/8, 5/12, 13/14, 15/100, 22/45, 27/35 is:<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Calculate the l.c.m of 7/10, 8/9, 3/8, 5/12, 13/14, 15/100, 22/45, 27/35 is:<|im_end|>
<|im_start|>assistant
<Z_272><ANSWER>

Question: Calculate the l.c.m of 7/10, 8/9, 3/8, 5/12, 13/14, 15/100, 22/45, 27/35 is:


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school has between 160 and 210 students enrolled. Each afternoon, all the students participate in a gym class divided into eight distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school has between 160 and 210 students enrolled. Each afternoon, all the students participate in a gym class divided into eight distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?<|im_end|>
<|im_start|>assistant
<Z_24><Z_343><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school has between 160 and 210 students enrolled. Each afternoon, all the students participate in a gym class divided into eight distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?<|im_end|>
<|im_start|>assistant
<Z_286><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A school has between 160 and 210 students enrolled. Each afternoon, all the students participate in a gym class divided into eight distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?<|im_end|>
<|im_start|>assistant
<Z_151><Z_340><ANSWER>

Question: A school has between 160 and 210 students enrolled. Each afternoon, all the students participate in a gym class divided into eight distinct sections. If one student is absent from school, the sections can all have the same number of students. What is the sum of all possible numbers of students enrolled at the school?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each assigned a random integer value, with $H = 12$. The value of a word comes from the sum of its letters' values. Given $MATH$ is 40 points, $TEAM$ is 50 points, and $MEET$ is 44 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each assigned a random integer value, with $H = 12$. The value of a word comes from the sum of its letters' values. Given $MATH$ is 40 points, $TEAM$ is 50 points, and $MEET$ is 44 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_217><Z_305><Z_68><Z_276><Z_420><Z_230><Z_322><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each assigned a random integer value, with $H = 12$. The value of a word comes from the sum of its letters' values. Given $MATH$ is 40 points, $TEAM$ is 50 points, and $MEET$ is 44 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_202><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The letters of the alphabet are each assigned a random integer value, with $H = 12$. The value of a word comes from the sum of its letters' values. Given $MATH$ is 40 points, $TEAM$ is 50 points, and $MEET$ is 44 points, what is the value of $A$?<|im_end|>
<|im_start|>assistant
<Z_417><Z_356><Z_240><Z_362><Z_170><Z_491><Z_180><Z_33><Z_388><ANSWER>

Question: The letters of the alphabet are each assigned a random integer value, with $H = 12$. The value of a word comes from the sum of its letters' values. Given $MATH$ is 40 points, $TEAM$ is 50 points, and $MEET$ is 44 points, what is the value of $A$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:

2019年1月1日后个人所得税税率表

| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |

Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.

A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:

2019年1月1日后个人所得税税率表

| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |

Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.

A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan.<|im_end|>
<|im_start|>assistant
<Z_477><Z_265><Z_127><Z_269><Z_494><Z_468><Z_413><Z_267><Z_415><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:

2019年1月1日后个人所得税税率表

| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |

Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.

A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan.<|im_end|>
<|im_start|>assistant
<Z_424><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:

2019年1月1日后个人所得税税率表

| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |

Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.

A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan.<|im_end|>
<|im_start|>assistant
<Z_279><Z_267><Z_470><Z_130><Z_33><Z_211><Z_204><Z_8><Z_241><ANSWER>

Question:
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:

2019年1月1日后个人所得税税率表

| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |

Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.

A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mary has a mystery book library. She started with 72 mystery books at the beginning of the year. She joined a book club that sent her 1 book each month of the year. She went to a bookstore halfway through the year and bought 5 more books, then bought some books at yard sales later. Her daughter gave her another book for her birthday, and her mother gave her a series of 4 books when Mary visited her. She donated 12 books she didn't want to read again to a charity shop and sold 3 to a used book store. She had 81 books at the end of the year. How many books did she buy at yard sales?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mary has a mystery book library. She started with 72 mystery books at the beginning of the year. She joined a book club that sent her 1 book each month of the year. She went to a bookstore halfway through the year and bought 5 more books, then bought some books at yard sales later. Her daughter gave her another book for her birthday, and her mother gave her a series of 4 books when Mary visited her. She donated 12 books she didn't want to read again to a charity shop and sold 3 to a used book store. She had 81 books at the end of the year. How many books did she buy at yard sales?<|im_end|>
<|im_start|>assistant
<Z_477><Z_371><Z_456><Z_376><Z_433><Z_341><Z_463><Z_460><Z_358><Z_181><Z_433><Z_308><Z_292><Z_147><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mary has a mystery book library. She started with 72 mystery books at the beginning of the year. She joined a book club that sent her 1 book each month of the year. She went to a bookstore halfway through the year and bought 5 more books, then bought some books at yard sales later. Her daughter gave her another book for her birthday, and her mother gave her a series of 4 books when Mary visited her. She donated 12 books she didn't want to read again to a charity shop and sold 3 to a used book store. She had 81 books at the end of the year. How many books did she buy at yard sales?<|im_end|>
<|im_start|>assistant
<Z_271><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mary has a mystery book library. She started with 72 mystery books at the beginning of the year. She joined a book club that sent her 1 book each month of the year. She went to a bookstore halfway through the year and bought 5 more books, then bought some books at yard sales later. Her daughter gave her another book for her birthday, and her mother gave her a series of 4 books when Mary visited her. She donated 12 books she didn't want to read again to a charity shop and sold 3 to a used book store. She had 81 books at the end of the year. How many books did she buy at yard sales?<|im_end|>
<|im_start|>assistant
<Z_14><Z_211><Z_44><Z_0><Z_170><Z_74><Z_0><Z_511><Z_85><Z_183><Z_477><Z_426><Z_48><Z_10><ANSWER>

Question: Mary has a mystery book library. She started with 72 mystery books at the beginning of the year. She joined a book club that sent her 1 book each month of the year. She went to a bookstore halfway through the year and bought 5 more books, then bought some books at yard sales later. Her daughter gave her another book for her birthday, and her mother gave her a series of 4 books when Mary visited her. She donated 12 books she didn't want to read again to a charity shop and sold 3 to a used book store. She had 81 books at the end of the year. How many books did she buy at yard sales?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(|a-b|=3\), \(|b-c|=4\), and \(|c-d|=5\). Additionally, \(a\) is fixed at 0. What is the sum of all possible values of \(|a-d|\)?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(|a-b|=3\), \(|b-c|=4\), and \(|c-d|=5\). Additionally, \(a\) is fixed at 0. What is the sum of all possible values of \(|a-d|\)?<|im_end|>
<|im_start|>assistant
<Z_369><Z_308><Z_311><Z_253><Z_381><Z_497><Z_67><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(|a-b|=3\), \(|b-c|=4\), and \(|c-d|=5\). Additionally, \(a\) is fixed at 0. What is the sum of all possible values of \(|a-d|\)?<|im_end|>
<|im_start|>assistant
<Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(|a-b|=3\), \(|b-c|=4\), and \(|c-d|=5\). Additionally, \(a\) is fixed at 0. What is the sum of all possible values of \(|a-d|\)?<|im_end|>
<|im_start|>assistant
<Z_296><Z_190><Z_203><Z_62><Z_337><Z_109><Z_285><ANSWER>

Question: Let \(a\), \(b\), \(c\), and \(d\) be real numbers such that \(|a-b|=3\), \(|b-c|=4\), and \(|c-d|=5\). Additionally, \(a\) is fixed at 0. What is the sum of all possible values of \(|a-d|\)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $g$ has the property that $g(4x-2) = x^2 - x + 2$ for all real numbers $x$. What is $g(6)$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $g$ has the property that $g(4x-2) = x^2 - x + 2$ for all real numbers $x$. What is $g(6)$?<|im_end|>
<|im_start|>assistant
<Z_369><Z_312><Z_343><Z_448><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $g$ has the property that $g(4x-2) = x^2 - x + 2$ for all real numbers $x$. What is $g(6)$?<|im_end|>
<|im_start|>assistant
<Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $g$ has the property that $g(4x-2) = x^2 - x + 2$ for all real numbers $x$. What is $g(6)$?<|im_end|>
<|im_start|>assistant
<Z_412><Z_14><Z_254><Z_230><ANSWER>

Question: A function $g$ has the property that $g(4x-2) = x^2 - x + 2$ for all real numbers $x$. What is $g(6)$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given vectors $\mathbf{a} = (n, -1)$, $\mathbf{b} = (-1, 1)$, $\mathbf{c} = (-1, 2)$, if $\mathbf{a} + \mathbf{b}$ is parallel to $\mathbf{c}$, then $n = \boxed{}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given vectors $\mathbf{a} = (n, -1)$, $\mathbf{b} = (-1, 1)$, $\mathbf{c} = (-1, 2)$, if $\mathbf{a} + \mathbf{b}$ is parallel to $\mathbf{c}$, then $n = \boxed{}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_100><Z_372><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given vectors $\mathbf{a} = (n, -1)$, $\mathbf{b} = (-1, 1)$, $\mathbf{c} = (-1, 2)$, if $\mathbf{a} + \mathbf{b}$ is parallel to $\mathbf{c}$, then $n = \boxed{}$.<|im_end|>
<|im_start|>assistant
<Z_494><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given vectors $\mathbf{a} = (n, -1)$, $\mathbf{b} = (-1, 1)$, $\mathbf{c} = (-1, 2)$, if $\mathbf{a} + \mathbf{b}$ is parallel to $\mathbf{c}$, then $n = \boxed{}$.<|im_end|>
<|im_start|>assistant
<Z_334><Z_82><Z_174><ANSWER>

Question: Given vectors $\mathbf{a} = (n, -1)$, $\mathbf{b} = (-1, 1)$, $\mathbf{c} = (-1, 2)$, if $\mathbf{a} + \mathbf{b}$ is parallel to $\mathbf{c}$, then $n = \boxed{}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two regular polygons have the same perimeter. If the first has 50 sides and a side length three times as long as the second, how many sides does the second have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two regular polygons have the same perimeter. If the first has 50 sides and a side length three times as long as the second, how many sides does the second have?<|im_end|>
<|im_start|>assistant
<Z_477><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two regular polygons have the same perimeter. If the first has 50 sides and a side length three times as long as the second, how many sides does the second have?<|im_end|>
<|im_start|>assistant
<Z_222><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Two regular polygons have the same perimeter. If the first has 50 sides and a side length three times as long as the second, how many sides does the second have?<|im_end|>
<|im_start|>assistant
<Z_343><Z_193><ANSWER>

Question: Two regular polygons have the same perimeter. If the first has 50 sides and a side length three times as long as the second, how many sides does the second have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?<|im_end|>
<|im_start|>assistant
<Z_369><Z_283><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?<|im_end|>
<|im_start|>assistant
<Z_83><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?<|im_end|>
<|im_start|>assistant
<Z_408><Z_387><Z_117><ANSWER>

Question: Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each permutation $a_1,a_2,a_3,\cdots,a_{12}$ of the integers $1,2,3,\cdots,12$, form the sum
\[|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}| + |a_{11} - a_{12}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each permutation $a_1,a_2,a_3,\cdots,a_{12}$ of the integers $1,2,3,\cdots,12$, form the sum
\[|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}| + |a_{11} - a_{12}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_213><Z_427><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each permutation $a_1,a_2,a_3,\cdots,a_{12}$ of the integers $1,2,3,\cdots,12$, form the sum
\[|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}| + |a_{11} - a_{12}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.<|im_end|>
<|im_start|>assistant
<Z_290><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For each permutation $a_1,a_2,a_3,\cdots,a_{12}$ of the integers $1,2,3,\cdots,12$, form the sum
\[|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}| + |a_{11} - a_{12}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.<|im_end|>
<|im_start|>assistant
<Z_431><Z_433><Z_135><ANSWER>

Question:
For each permutation $a_1,a_2,a_3,\cdots,a_{12}$ of the integers $1,2,3,\cdots,12$, form the sum
\[|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}| + |a_{11} - a_{12}|.\]
The average value of all such sums can be written in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*<|im_end|>
<|im_start|>assistant
<Z_477><Z_231><Z_488><Z_276><Z_412><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*<|im_end|>
<|im_start|>assistant
<Z_510><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*<|im_end|>
<|im_start|>assistant
<Z_2><Z_140><Z_81><Z_294><Z_104><ANSWER>

Question:
We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).<|im_end|>
<|im_start|>assistant
<Z_111><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).<|im_end|>
<|im_start|>assistant
<Z_438><ANSWER>

Question: Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?<|im_end|>
<|im_start|>assistant
<Z_369><Z_130><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?<|im_end|>
<|im_start|>assistant
<Z_385><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?<|im_end|>
<|im_start|>assistant
<Z_314><Z_84><ANSWER>

Question: For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain integer has $4$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain integer has $4$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_205><Z_400><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain integer has $4$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?<|im_end|>
<|im_start|>assistant
<Z_325><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain integer has $4$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?<|im_end|>
<|im_start|>assistant
<Z_47><Z_247><Z_503><ANSWER>

Question: A certain integer has $4$ digits when written in base $7$. The same integer has $d$ digits when written in base $3$. What is the sum of all possible values of $d$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $2024$ and $2295$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $2024$ and $2295$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_400><Z_428><Z_361><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $2024$ and $2295$?<|im_end|>
<|im_start|>assistant
<Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the greatest common divisor of $2024$ and $2295$?<|im_end|>
<|im_start|>assistant
<Z_242><Z_232><Z_375><Z_395><ANSWER>

Question: What is the greatest common divisor of $2024$ and $2295$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the simple interest on a sum of money will be rs . 800 after 10 years . if the principal is trebled after 5 years what will be the total interest at the end of the tenth year ?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the simple interest on a sum of money will be rs . 800 after 10 years . if the principal is trebled after 5 years what will be the total interest at the end of the tenth year ?<|im_end|>
<|im_start|>assistant
<Z_369><Z_218><Z_260><Z_99><Z_139><Z_230><Z_433><Z_320><Z_415><Z_446><Z_309><Z_111><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the simple interest on a sum of money will be rs . 800 after 10 years . if the principal is trebled after 5 years what will be the total interest at the end of the tenth year ?<|im_end|>
<|im_start|>assistant
<Z_210><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the simple interest on a sum of money will be rs . 800 after 10 years . if the principal is trebled after 5 years what will be the total interest at the end of the tenth year ?<|im_end|>
<|im_start|>assistant
<Z_250><Z_484><Z_312><Z_156><Z_168><Z_318><Z_42><Z_440><Z_380><Z_162><Z_193><Z_322><ANSWER>

Question: the simple interest on a sum of money will be rs . 800 after 10 years . if the principal is trebled after 5 years what will be the total interest at the end of the tenth year ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 360-meter long train crosses a man standing on the platform in a certain amount of time. The speed of the train is 216 km/h. How many seconds does it take for the train to cross the man?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 360-meter long train crosses a man standing on the platform in a certain amount of time. The speed of the train is 216 km/h. How many seconds does it take for the train to cross the man?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 360-meter long train crosses a man standing on the platform in a certain amount of time. The speed of the train is 216 km/h. How many seconds does it take for the train to cross the man?<|im_end|>
<|im_start|>assistant
<Z_201><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A 360-meter long train crosses a man standing on the platform in a certain amount of time. The speed of the train is 216 km/h. How many seconds does it take for the train to cross the man?<|im_end|>
<|im_start|>assistant
<Z_26><ANSWER>

Question: A 360-meter long train crosses a man standing on the platform in a certain amount of time. The speed of the train is 216 km/h. How many seconds does it take for the train to cross the man?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rob, Jess, and Alex are friends who collect baseball cards, and each of them has a collection that includes some doubles. One third of Rob's cards are doubles, while one fourth of Alex's cards are doubles. Moreover, Jess has 5 times as many doubles as Rob.

In total, Jess has 40 doubles baseball cards. Comparing the ratio of doubles between all three friends, determine how many unique baseball cards Rob, Jess, and Alex have together in their collections.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rob, Jess, and Alex are friends who collect baseball cards, and each of them has a collection that includes some doubles. One third of Rob's cards are doubles, while one fourth of Alex's cards are doubles. Moreover, Jess has 5 times as many doubles as Rob.

In total, Jess has 40 doubles baseball cards. Comparing the ratio of doubles between all three friends, determine how many unique baseball cards Rob, Jess, and Alex have together in their collections.<|im_end|>
<|im_start|>assistant
<Z_369><Z_421><Z_416><Z_490><Z_194><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rob, Jess, and Alex are friends who collect baseball cards, and each of them has a collection that includes some doubles. One third of Rob's cards are doubles, while one fourth of Alex's cards are doubles. Moreover, Jess has 5 times as many doubles as Rob.

In total, Jess has 40 doubles baseball cards. Comparing the ratio of doubles between all three friends, determine how many unique baseball cards Rob, Jess, and Alex have together in their collections.<|im_end|>
<|im_start|>assistant
<Z_482><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rob, Jess, and Alex are friends who collect baseball cards, and each of them has a collection that includes some doubles. One third of Rob's cards are doubles, while one fourth of Alex's cards are doubles. Moreover, Jess has 5 times as many doubles as Rob.

In total, Jess has 40 doubles baseball cards. Comparing the ratio of doubles between all three friends, determine how many unique baseball cards Rob, Jess, and Alex have together in their collections.<|im_end|>
<|im_start|>assistant
<Z_10><Z_182><Z_439><Z_64><Z_106><ANSWER>

Question:
Rob, Jess, and Alex are friends who collect baseball cards, and each of them has a collection that includes some doubles. One third of Rob's cards are doubles, while one fourth of Alex's cards are doubles. Moreover, Jess has 5 times as many doubles as Rob.

In total, Jess has 40 doubles baseball cards. Comparing the ratio of doubles between all three friends, determine how many unique baseball cards Rob, Jess, and Alex have together in their collections.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 300 meters long is running at a speed of 35 km/hour. It takes 42.68571428571429 seconds to pass a bridge of a certain length. What is the length of the bridge in meters?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 300 meters long is running at a speed of 35 km/hour. It takes 42.68571428571429 seconds to pass a bridge of a certain length. What is the length of the bridge in meters?<|im_end|>
<|im_start|>assistant
<Z_24><Z_322><Z_393><Z_185><Z_366><Z_448><Z_450><Z_497><Z_287><Z_99><Z_497><Z_413><Z_386><Z_169><Z_83><Z_200><Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 300 meters long is running at a speed of 35 km/hour. It takes 42.68571428571429 seconds to pass a bridge of a certain length. What is the length of the bridge in meters?<|im_end|>
<|im_start|>assistant
<Z_8><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train that is 300 meters long is running at a speed of 35 km/hour. It takes 42.68571428571429 seconds to pass a bridge of a certain length. What is the length of the bridge in meters?<|im_end|>
<|im_start|>assistant
<Z_76><Z_310><Z_510><Z_314><Z_430><Z_139><Z_442><Z_387><Z_388><Z_472><Z_235><Z_227><Z_454><Z_331><Z_31><Z_212><Z_362><ANSWER>

Question: A train that is 300 meters long is running at a speed of 35 km/hour. It takes 42.68571428571429 seconds to pass a bridge of a certain length. What is the length of the bridge in meters?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\).<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\).<|im_end|>
<|im_start|>assistant
<Z_477><Z_393><Z_205><Z_162><Z_293><Z_166><Z_319><Z_223><Z_470><Z_243><Z_260><Z_220><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\).<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\).<|im_end|>
<|im_start|>assistant
<Z_155><Z_25><Z_465><Z_49><Z_2><Z_488><Z_177><Z_384><Z_16><Z_302><Z_109><Z_185><ANSWER>

Question: Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
85% of a certain number is greater than 4/5 of 25 by 14. What is that number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
85% of a certain number is greater than 4/5 of 25 by 14. What is that number?<|im_end|>
<|im_start|>assistant
<Z_24><Z_230><Z_98><Z_175><Z_439><Z_185><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
85% of a certain number is greater than 4/5 of 25 by 14. What is that number?<|im_end|>
<|im_start|>assistant
<Z_488><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
85% of a certain number is greater than 4/5 of 25 by 14. What is that number?<|im_end|>
<|im_start|>assistant
<Z_430><Z_338><Z_138><Z_312><Z_118><Z_330><ANSWER>

Question: 85% of a certain number is greater than 4/5 of 25 by 14. What is that number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Vanessa and her mom were picking carrots from their garden. Vanessa picked 17 carrots and her mother picked some. If only 24 of the carrots were good, and they had 7 bad carrots, how many carrots did Vanessa's mother pick?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Vanessa and her mom were picking carrots from their garden. Vanessa picked 17 carrots and her mother picked some. If only 24 of the carrots were good, and they had 7 bad carrots, how many carrots did Vanessa's mother pick?<|im_end|>
<|im_start|>assistant
<Z_477><Z_309><Z_381><Z_309><Z_368><Z_426><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Vanessa and her mom were picking carrots from their garden. Vanessa picked 17 carrots and her mother picked some. If only 24 of the carrots were good, and they had 7 bad carrots, how many carrots did Vanessa's mother pick?<|im_end|>
<|im_start|>assistant
<Z_222><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Vanessa and her mom were picking carrots from their garden. Vanessa picked 17 carrots and her mother picked some. If only 24 of the carrots were good, and they had 7 bad carrots, how many carrots did Vanessa's mother pick?<|im_end|>
<|im_start|>assistant
<Z_501><Z_475><Z_236><Z_238><Z_36><Z_389><Z_1><ANSWER>

Question: Vanessa and her mom were picking carrots from their garden. Vanessa picked 17 carrots and her mother picked some. If only 24 of the carrots were good, and they had 7 bad carrots, how many carrots did Vanessa's mother pick?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system \(xOy\), given points \(A(0,0)\), \(B(3,0)\), and \(C(2,2)\). Suppose the transformations \(T_1\) and \(T_2\) correspond to matrices \(M= \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\) and \(N= \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\), respectively. Find the area of the figure obtained by applying transformations \(T_1\) and \(T_2\) to \(\triangle ABC\) in that order.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system \(xOy\), given points \(A(0,0)\), \(B(3,0)\), and \(C(2,2)\). Suppose the transformations \(T_1\) and \(T_2\) correspond to matrices \(M= \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\) and \(N= \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\), respectively. Find the area of the figure obtained by applying transformations \(T_1\) and \(T_2\) to \(\triangle ABC\) in that order.<|im_end|>
<|im_start|>assistant
<Z_477><Z_165><Z_330><Z_213><Z_68><Z_87><Z_178><Z_118><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system \(xOy\), given points \(A(0,0)\), \(B(3,0)\), and \(C(2,2)\). Suppose the transformations \(T_1\) and \(T_2\) correspond to matrices \(M= \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\) and \(N= \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\), respectively. Find the area of the figure obtained by applying transformations \(T_1\) and \(T_2\) to \(\triangle ABC\) in that order.<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the Cartesian coordinate system \(xOy\), given points \(A(0,0)\), \(B(3,0)\), and \(C(2,2)\). Suppose the transformations \(T_1\) and \(T_2\) correspond to matrices \(M= \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\) and \(N= \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\), respectively. Find the area of the figure obtained by applying transformations \(T_1\) and \(T_2\) to \(\triangle ABC\) in that order.<|im_end|>
<|im_start|>assistant
<Z_393><Z_29><Z_384><Z_334><Z_254><Z_240><Z_410><Z_208><ANSWER>

Question: In the Cartesian coordinate system \(xOy\), given points \(A(0,0)\), \(B(3,0)\), and \(C(2,2)\). Suppose the transformations \(T_1\) and \(T_2\) correspond to matrices \(M= \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\) and \(N= \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\), respectively. Find the area of the figure obtained by applying transformations \(T_1\) and \(T_2\) to \(\triangle ABC\) in that order.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 2023 positive numbers with both their sum and the sum of their reciprocals equal to 2024. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 2023 positive numbers with both their sum and the sum of their reciprocals equal to 2024. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_200><Z_236><Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 2023 positive numbers with both their sum and the sum of their reciprocals equal to 2024. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}$.<|im_end|>
<|im_start|>assistant
<Z_440><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 2023 positive numbers with both their sum and the sum of their reciprocals equal to 2024. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}$.<|im_end|>
<|im_start|>assistant
<Z_281><Z_252><Z_307><Z_289><ANSWER>

Question: There are 2023 positive numbers with both their sum and the sum of their reciprocals equal to 2024. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average monthly salary of 18 employees in an organisation is Rs. 2000. If the manager's salary is added, then the average salary increases by Rs. 200. What is the manager's monthly salary?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average monthly salary of 18 employees in an organisation is Rs. 2000. If the manager's salary is added, then the average salary increases by Rs. 200. What is the manager's monthly salary?<|im_end|>
<|im_start|>assistant
<Z_477><Z_236><Z_155><Z_283><Z_340><Z_117><Z_468><Z_87><Z_54><Z_241><Z_352><Z_494><Z_228><Z_403><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average monthly salary of 18 employees in an organisation is Rs. 2000. If the manager's salary is added, then the average salary increases by Rs. 200. What is the manager's monthly salary?<|im_end|>
<|im_start|>assistant
<Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average monthly salary of 18 employees in an organisation is Rs. 2000. If the manager's salary is added, then the average salary increases by Rs. 200. What is the manager's monthly salary?<|im_end|>
<|im_start|>assistant
<Z_38><Z_333><Z_16><Z_243><Z_74><Z_116><Z_105><Z_51><Z_338><Z_166><Z_41><Z_303><Z_361><Z_379><ANSWER>

Question: The average monthly salary of 18 employees in an organisation is Rs. 2000. If the manager's salary is added, then the average salary increases by Rs. 200. What is the manager's monthly salary?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a , b and c rent a pasture . if a puts 10 oxen for 7 months , b puts 12 oxen for 5 months and c puts 15 oxen for 3 months for grazing and the rent of the pasture is rs . 105 , then how much amount should c pay as his share of rent ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a , b and c rent a pasture . if a puts 10 oxen for 7 months , b puts 12 oxen for 5 months and c puts 15 oxen for 3 months for grazing and the rent of the pasture is rs . 105 , then how much amount should c pay as his share of rent ?<|im_end|>
<|im_start|>assistant
<Z_468><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a , b and c rent a pasture . if a puts 10 oxen for 7 months , b puts 12 oxen for 5 months and c puts 15 oxen for 3 months for grazing and the rent of the pasture is rs . 105 , then how much amount should c pay as his share of rent ?<|im_end|>
<|im_start|>assistant
<Z_295><ANSWER>

Question: a , b and c rent a pasture . if a puts 10 oxen for 7 months , b puts 12 oxen for 5 months and c puts 15 oxen for 3 months for grazing and the rent of the pasture is rs . 105 , then how much amount should c pay as his share of rent ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maggie bought some packs of red bouncy balls, 8 packs of yellow bouncy balls, and 4 packs of green bouncy balls. There were 10 bouncy balls in each package. Maggie bought 160 bouncy balls in all. How many packs of red bouncy balls did Maggie buy?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maggie bought some packs of red bouncy balls, 8 packs of yellow bouncy balls, and 4 packs of green bouncy balls. There were 10 bouncy balls in each package. Maggie bought 160 bouncy balls in all. How many packs of red bouncy balls did Maggie buy?<|im_end|>
<|im_start|>assistant
<Z_477><Z_351><Z_376><Z_71><Z_331><Z_193><Z_59><Z_286><Z_246><Z_421><Z_434><Z_413><Z_181><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maggie bought some packs of red bouncy balls, 8 packs of yellow bouncy balls, and 4 packs of green bouncy balls. There were 10 bouncy balls in each package. Maggie bought 160 bouncy balls in all. How many packs of red bouncy balls did Maggie buy?<|im_end|>
<|im_start|>assistant
<Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maggie bought some packs of red bouncy balls, 8 packs of yellow bouncy balls, and 4 packs of green bouncy balls. There were 10 bouncy balls in each package. Maggie bought 160 bouncy balls in all. How many packs of red bouncy balls did Maggie buy?<|im_end|>
<|im_start|>assistant
<Z_68><Z_352><Z_138><Z_341><Z_482><Z_449><Z_351><Z_8><Z_408><Z_426><Z_186><Z_200><Z_497><ANSWER>

Question: Maggie bought some packs of red bouncy balls, 8 packs of yellow bouncy balls, and 4 packs of green bouncy balls. There were 10 bouncy balls in each package. Maggie bought 160 bouncy balls in all. How many packs of red bouncy balls did Maggie buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yoojeong gave Eunji 8 of the marbles she had at first, and there are 24 left. How many marbles did Yoojeong have before giving them to Eunji?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yoojeong gave Eunji 8 of the marbles she had at first, and there are 24 left. How many marbles did Yoojeong have before giving them to Eunji?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yoojeong gave Eunji 8 of the marbles she had at first, and there are 24 left. How many marbles did Yoojeong have before giving them to Eunji?<|im_end|>
<|im_start|>assistant
<Z_235><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Yoojeong gave Eunji 8 of the marbles she had at first, and there are 24 left. How many marbles did Yoojeong have before giving them to Eunji?<|im_end|>
<|im_start|>assistant
<Z_166><ANSWER>

Question: Yoojeong gave Eunji 8 of the marbles she had at first, and there are 24 left. How many marbles did Yoojeong have before giving them to Eunji?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 animals on a farm. Out of the 15 animals, 6 are ducks, 5 are dogs, 3 are spiders, and 1 is a three-legged dog. How many animal legs are there in total on the farm?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 animals on a farm. Out of the 15 animals, 6 are ducks, 5 are dogs, 3 are spiders, and 1 is a three-legged dog. How many animal legs are there in total on the farm?<|im_end|>
<|im_start|>assistant
<Z_369><Z_366><Z_360><Z_437><Z_497><Z_175><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 animals on a farm. Out of the 15 animals, 6 are ducks, 5 are dogs, 3 are spiders, and 1 is a three-legged dog. How many animal legs are there in total on the farm?<|im_end|>
<|im_start|>assistant
<Z_128><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 15 animals on a farm. Out of the 15 animals, 6 are ducks, 5 are dogs, 3 are spiders, and 1 is a three-legged dog. How many animal legs are there in total on the farm?<|im_end|>
<|im_start|>assistant
<Z_304><Z_208><Z_439><Z_294><Z_110><Z_91><ANSWER>

Question: There are 15 animals on a farm. Out of the 15 animals, 6 are ducks, 5 are dogs, 3 are spiders, and 1 is a three-legged dog. How many animal legs are there in total on the farm?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four squirrels together ate a total of 2020 nuts, with each squirrel eating at least 103 nuts. The first squirrel ate more nuts than any of the other three squirrels. The second and third squirrels together ate 1277 nuts.

How many nuts did the first squirrel eat?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four squirrels together ate a total of 2020 nuts, with each squirrel eating at least 103 nuts. The first squirrel ate more nuts than any of the other three squirrels. The second and third squirrels together ate 1277 nuts.

How many nuts did the first squirrel eat?<|im_end|>
<|im_start|>assistant
<Z_477><Z_509><Z_267><Z_319><Z_460><Z_313><Z_304><Z_449><Z_310><Z_443><Z_295><Z_176><Z_414><Z_267><Z_184><Z_393><Z_443><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four squirrels together ate a total of 2020 nuts, with each squirrel eating at least 103 nuts. The first squirrel ate more nuts than any of the other three squirrels. The second and third squirrels together ate 1277 nuts.

How many nuts did the first squirrel eat?<|im_end|>
<|im_start|>assistant
<Z_421><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Four squirrels together ate a total of 2020 nuts, with each squirrel eating at least 103 nuts. The first squirrel ate more nuts than any of the other three squirrels. The second and third squirrels together ate 1277 nuts.

How many nuts did the first squirrel eat?<|im_end|>
<|im_start|>assistant
<Z_69><Z_363><Z_436><Z_228><Z_401><Z_145><Z_275><Z_20><Z_37><Z_193><Z_248><Z_118><Z_145><Z_494><Z_61><Z_261><Z_260><Z_184><ANSWER>

Question:
Four squirrels together ate a total of 2020 nuts, with each squirrel eating at least 103 nuts. The first squirrel ate more nuts than any of the other three squirrels. The second and third squirrels together ate 1277 nuts.

How many nuts did the first squirrel eat?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of the red string is 8 meters (m). If the blue string is 8 times the length of the white string and the white string is 5 times the length of the red string, how many meters (m) is the blue string?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of the red string is 8 meters (m). If the blue string is 8 times the length of the white string and the white string is 5 times the length of the red string, how many meters (m) is the blue string?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of the red string is 8 meters (m). If the blue string is 8 times the length of the white string and the white string is 5 times the length of the red string, how many meters (m) is the blue string?<|im_end|>
<|im_start|>assistant
<Z_257><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The length of the red string is 8 meters (m). If the blue string is 8 times the length of the white string and the white string is 5 times the length of the red string, how many meters (m) is the blue string?<|im_end|>
<|im_start|>assistant
<Z_449><ANSWER>

Question: The length of the red string is 8 meters (m). If the blue string is 8 times the length of the white string and the white string is 5 times the length of the red string, how many meters (m) is the blue string?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tricia is some fraction of Amilia's age and Amilia is a quarter of Yorick's age. Yorick is twice Eugene's age and Khloe is a third of Eugene's age. Rupert is 10 years older than Khloe but 2 years younger than Vincent who is 22 years old. How old, in years, is Tricia if she is 5 years old?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tricia is some fraction of Amilia's age and Amilia is a quarter of Yorick's age. Yorick is twice Eugene's age and Khloe is a third of Eugene's age. Rupert is 10 years older than Khloe but 2 years younger than Vincent who is 22 years old. How old, in years, is Tricia if she is 5 years old?<|im_end|>
<|im_start|>assistant
<Z_477><Z_273><Z_219><Z_67><Z_448><Z_111><Z_195><Z_350><Z_366><Z_473><Z_331><Z_254><Z_404><Z_269><Z_211><Z_345><Z_320><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tricia is some fraction of Amilia's age and Amilia is a quarter of Yorick's age. Yorick is twice Eugene's age and Khloe is a third of Eugene's age. Rupert is 10 years older than Khloe but 2 years younger than Vincent who is 22 years old. How old, in years, is Tricia if she is 5 years old?<|im_end|>
<|im_start|>assistant
<Z_19><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tricia is some fraction of Amilia's age and Amilia is a quarter of Yorick's age. Yorick is twice Eugene's age and Khloe is a third of Eugene's age. Rupert is 10 years older than Khloe but 2 years younger than Vincent who is 22 years old. How old, in years, is Tricia if she is 5 years old?<|im_end|>
<|im_start|>assistant
<Z_38><Z_190><Z_382><Z_143><Z_71><Z_294><Z_270><Z_213><Z_451><Z_313><Z_237><Z_97><Z_102><Z_100><Z_12><Z_284><Z_274><ANSWER>

Question: Tricia is some fraction of Amilia's age and Amilia is a quarter of Yorick's age. Yorick is twice Eugene's age and Khloe is a third of Eugene's age. Rupert is 10 years older than Khloe but 2 years younger than Vincent who is 22 years old. How old, in years, is Tricia if she is 5 years old?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_157><Z_50><Z_450><Z_388><Z_439><Z_251><Z_191><Z_205><Z_71><Z_394><Z_219><Z_396><Z_285><Z_475><Z_322><Z_255><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.<|im_end|>
<|im_start|>assistant
<Z_300><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.<|im_end|>
<|im_start|>assistant
<Z_79><Z_273><Z_318><Z_72><Z_387><Z_445><Z_7><Z_62><Z_442><Z_395><Z_369><Z_240><Z_443><Z_450><Z_6><Z_90><Z_379><ANSWER>

Question: Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jaylen has 5 carrots and some cucumbers. Jaylen has twice as many bell peppers as Kristin. Jaylen has 3 less than half as many green beans as Kristin. If Kristin has 2 bell peppers and 20 green beans, and Jaylen has 18 vegetables in total, how many cucumbers does Jaylen have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jaylen has 5 carrots and some cucumbers. Jaylen has twice as many bell peppers as Kristin. Jaylen has 3 less than half as many green beans as Kristin. If Kristin has 2 bell peppers and 20 green beans, and Jaylen has 18 vegetables in total, how many cucumbers does Jaylen have?<|im_end|>
<|im_start|>assistant
<Z_24><Z_176><Z_430><Z_162><Z_196><Z_373><Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jaylen has 5 carrots and some cucumbers. Jaylen has twice as many bell peppers as Kristin. Jaylen has 3 less than half as many green beans as Kristin. If Kristin has 2 bell peppers and 20 green beans, and Jaylen has 18 vegetables in total, how many cucumbers does Jaylen have?<|im_end|>
<|im_start|>assistant
<Z_382><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jaylen has 5 carrots and some cucumbers. Jaylen has twice as many bell peppers as Kristin. Jaylen has 3 less than half as many green beans as Kristin. If Kristin has 2 bell peppers and 20 green beans, and Jaylen has 18 vegetables in total, how many cucumbers does Jaylen have?<|im_end|>
<|im_start|>assistant
<Z_250><Z_294><Z_155><Z_16><Z_127><Z_9><Z_51><ANSWER>

Question: Jaylen has 5 carrots and some cucumbers. Jaylen has twice as many bell peppers as Kristin. Jaylen has 3 less than half as many green beans as Kristin. If Kristin has 2 bell peppers and 20 green beans, and Jaylen has 18 vegetables in total, how many cucumbers does Jaylen have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sebastian works everyday and gets his salary of 1043 dollars on a weekly basis. If Sebastian didn't go to work on some days this week, his deducted salary was 745 dollars. How many days did he not go to work?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sebastian works everyday and gets his salary of 1043 dollars on a weekly basis. If Sebastian didn't go to work on some days this week, his deducted salary was 745 dollars. How many days did he not go to work?<|im_end|>
<|im_start|>assistant
<Z_24><Z_185><Z_400><Z_219><Z_487><Z_410><Z_450><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sebastian works everyday and gets his salary of 1043 dollars on a weekly basis. If Sebastian didn't go to work on some days this week, his deducted salary was 745 dollars. How many days did he not go to work?<|im_end|>
<|im_start|>assistant
<Z_145><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Sebastian works everyday and gets his salary of 1043 dollars on a weekly basis. If Sebastian didn't go to work on some days this week, his deducted salary was 745 dollars. How many days did he not go to work?<|im_end|>
<|im_start|>assistant
<Z_0><Z_183><Z_357><Z_496><Z_449><Z_444><Z_309><ANSWER>

Question: Sebastian works everyday and gets his salary of 1043 dollars on a weekly basis. If Sebastian didn't go to work on some days this week, his deducted salary was 745 dollars. How many days did he not go to work?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\dbinom{20}{6}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\dbinom{20}{6}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_134><Z_418><Z_244><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\dbinom{20}{6}$.<|im_end|>
<|im_start|>assistant
<Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute $\dbinom{20}{6}$.<|im_end|>
<|im_start|>assistant
<Z_135><Z_510><Z_129><Z_216><ANSWER>

Question: Compute $\dbinom{20}{6}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a new operation, observe the following expressions:
$1\odot 3=1\times 2+3=5$, $4\odot \left(-1\right)=4\times 2-1=7$;
$(-2)\odot 3=\left(-2\right)\times 2+3=-1$; $6\odot 5=6\times 2+5=17$; $\ldots$
$(1)$ Please think about: Express $x\odot y$ with an algebraic expression as ______;
$(2)$ If $a\odot \left(-6b\right)=4$, calculate the value of $\left(a-5b\right)\odot \left(a+b\right)$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a new operation, observe the following expressions:
$1\odot 3=1\times 2+3=5$, $4\odot \left(-1\right)=4\times 2-1=7$;
$(-2)\odot 3=\left(-2\right)\times 2+3=-1$; $6\odot 5=6\times 2+5=17$; $\ldots$
$(1)$ Please think about: Express $x\odot y$ with an algebraic expression as ______;
$(2)$ If $a\odot \left(-6b\right)=4$, calculate the value of $\left(a-5b\right)\odot \left(a+b\right)$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_395><Z_327><Z_491><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a new operation, observe the following expressions:
$1\odot 3=1\times 2+3=5$, $4\odot \left(-1\right)=4\times 2-1=7$;
$(-2)\odot 3=\left(-2\right)\times 2+3=-1$; $6\odot 5=6\times 2+5=17$; $\ldots$
$(1)$ Please think about: Express $x\odot y$ with an algebraic expression as ______;
$(2)$ If $a\odot \left(-6b\right)=4$, calculate the value of $\left(a-5b\right)\odot \left(a+b\right)$.<|im_end|>
<|im_start|>assistant
<Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define a new operation, observe the following expressions:
$1\odot 3=1\times 2+3=5$, $4\odot \left(-1\right)=4\times 2-1=7$;
$(-2)\odot 3=\left(-2\right)\times 2+3=-1$; $6\odot 5=6\times 2+5=17$; $\ldots$
$(1)$ Please think about: Express $x\odot y$ with an algebraic expression as ______;
$(2)$ If $a\odot \left(-6b\right)=4$, calculate the value of $\left(a-5b\right)\odot \left(a+b\right)$.<|im_end|>
<|im_start|>assistant
<Z_123><Z_105><Z_182><Z_152><Z_317><ANSWER>

Question:
Define a new operation, observe the following expressions:
$1\odot 3=1\times 2+3=5$, $4\odot \left(-1\right)=4\times 2-1=7$;
$(-2)\odot 3=\left(-2\right)\times 2+3=-1$; $6\odot 5=6\times 2+5=17$; $\ldots$
$(1)$ Please think about: Express $x\odot y$ with an algebraic expression as ______;
$(2)$ If $a\odot \left(-6b\right)=4$, calculate the value of $\left(a-5b\right)\odot \left(a+b\right)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} 4 \\ 10 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} 4 \\ 10 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_143><Z_393><Z_195><Z_94><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} 4 \\ 10 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2$.<|im_end|>
<|im_start|>assistant
<Z_243><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} 4 \\ 10 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2$.<|im_end|>
<|im_start|>assistant
<Z_161><Z_109><Z_391><Z_391><Z_442><ANSWER>

Question: Let $\mathbf{a}$ and $\mathbf{b}$ be vectors, and let $\mathbf{m}$ be the midpoint of $\mathbf{a}$ and $\mathbf{b}$. Given $\mathbf{m} = \begin{pmatrix} 4 \\ 10 \end{pmatrix}$ and $\mathbf{a} \cdot \mathbf{b} = 10$, find $\|\mathbf{a}\|^2 + \|\mathbf{b}\|^2$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michelle uses 40 beads in making a necklace. She uses 5 blue beads and 2 times as many red beads as blue beads. Then she uses the same number of white beads as the blue and red beads combined. If the rest of the beads she uses are silver, how many silver beads does she use?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michelle uses 40 beads in making a necklace. She uses 5 blue beads and 2 times as many red beads as blue beads. Then she uses the same number of white beads as the blue and red beads combined. If the rest of the beads she uses are silver, how many silver beads does she use?<|im_end|>
<|im_start|>assistant
<Z_477><Z_350><Z_274><Z_465><Z_39><Z_312><Z_491><Z_159><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michelle uses 40 beads in making a necklace. She uses 5 blue beads and 2 times as many red beads as blue beads. Then she uses the same number of white beads as the blue and red beads combined. If the rest of the beads she uses are silver, how many silver beads does she use?<|im_end|>
<|im_start|>assistant
<Z_365><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Michelle uses 40 beads in making a necklace. She uses 5 blue beads and 2 times as many red beads as blue beads. Then she uses the same number of white beads as the blue and red beads combined. If the rest of the beads she uses are silver, how many silver beads does she use?<|im_end|>
<|im_start|>assistant
<Z_453><Z_348><Z_46><Z_55><Z_258><Z_310><Z_500><Z_270><ANSWER>

Question: Michelle uses 40 beads in making a necklace. She uses 5 blue beads and 2 times as many red beads as blue beads. Then she uses the same number of white beads as the blue and red beads combined. If the rest of the beads she uses are silver, how many silver beads does she use?