<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_104><Z_27><Z_36><Z_93><Z_40><Z_41><Z_39><Z_41><Z_80><Z_126><Z_60><Z_45><Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?<|im_end|>
<|im_start|>assistant
<Z_122><Z_36><Z_107><Z_3><Z_0><Z_105><Z_120><Z_59><Z_90><Z_105><Z_117><Z_127><Z_39><ANSWER>

Question: In Arun's opinion, his weight is greater than 62 kg but less than some value. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that his weight cannot be greater than 65 kg. If all of them are correct in their estimation, the average of different probable weights of Arun is 64 kg. What is the upper limit of Arun's weight according to his own opinion?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.<|im_end|>
<|im_start|>assistant
<Z_74><ANSWER>

Question:
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot 9 = 63$. This is a contradiction. The set cannot consist of 9 numbers.

b) Let the number of numbers in the set be $2n+1$ (where $n$ is natural). In the set, there will be exactly $n+1$ numbers, each of which does not exceed the median, which is 2. The remaining $n$ numbers do not exceed 13. Therefore, the sum of all the numbers in the set is at most:
$$
13n + 2(n + 1) = 15n + 2
$$
On the other hand, this sum is equal to $7(2n+1) = 14n+7$. From the inequality $14n+7 \leq 15n+2$, we get that $n \geq 5$. Consequently, the set contains at least $2 \cdot 5 + 1 = 11$ numbers. Here is an example to show that this is possible. The set
$$
2, 2, 2, 2, 2, 2, 13, 13, 13, 13, 13
$$
consists of 11 numbers and satisfies the conditions.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers less than 1000 that are cubes of perfect squares?<|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
What is the sum of all positive integers less than 1000 that are cubes of perfect squares?<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all positive integers less than 1000 that are cubes of perfect squares?<|im_end|>
<|im_start|>assistant
<Z_55><ANSWER>

Question: What is the sum of all positive integers less than 1000 that are cubes of perfect squares?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?<|im_end|>
<|im_start|>assistant
<Z_104><Z_73><Z_31><Z_73><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?<|im_end|>
<|im_start|>assistant
<Z_16><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?<|im_end|>
<|im_start|>assistant
<Z_48><Z_87><Z_113><Z_103><ANSWER>

Question: An airplane has four sections: First Class (36 seats), Business Class ($30\%$ of the total number of seats), Economy ($\frac{3}{5}$ of the total number of seats), and Premium Economy (remaining seats). How many seats does the airplane have in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
\( n \) is a positive integer. If \( 2n \) has 28 positive factors and \( 3n \) has 30 positive factors, then the number of positive factors of \( 6n \) is:
A. 32
B. 34
C. 35
D. 36
E. 38<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
\( n \) is a positive integer. If \( 2n \) has 28 positive factors and \( 3n \) has 30 positive factors, then the number of positive factors of \( 6n \) is:
A. 32
B. 34
C. 35
D. 36
E. 38<|im_end|>
<|im_start|>assistant
<Z_94><Z_86><Z_13><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
\( n \) is a positive integer. If \( 2n \) has 28 positive factors and \( 3n \) has 30 positive factors, then the number of positive factors of \( 6n \) is:
A. 32
B. 34
C. 35
D. 36
E. 38<|im_end|>
<|im_start|>assistant
<Z_122><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
\( n \) is a positive integer. If \( 2n \) has 28 positive factors and \( 3n \) has 30 positive factors, then the number of positive factors of \( 6n \) is:
A. 32
B. 34
C. 35
D. 36
E. 38<|im_end|>
<|im_start|>assistant
<Z_98><Z_64><Z_108><ANSWER>

Question:
\( n \) is a positive integer. If \( 2n \) has 28 positive factors and \( 3n \) has 30 positive factors, then the number of positive factors of \( 6n \) is:
A. 32
B. 34
C. 35
D. 36
E. 38


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ruth prepared sandwiches. She ate 1 sandwich and gave 2 sandwiches to her brother. Her first cousin arrived and ate 2 sandwiches. Then her two other cousins arrived and ate 1 sandwich each. There were 3 sandwiches left. How many sandwiches did Ruth prepare?<|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
Ruth prepared sandwiches. She ate 1 sandwich and gave 2 sandwiches to her brother. Her first cousin arrived and ate 2 sandwiches. Then her two other cousins arrived and ate 1 sandwich each. There were 3 sandwiches left. How many sandwiches did Ruth prepare?<|im_end|>
<|im_start|>assistant
<Z_57><Z_103><Z_66><Z_12><Z_47><Z_115><Z_47><Z_114><Z_39><Z_96><Z_67><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ruth prepared sandwiches. She ate 1 sandwich and gave 2 sandwiches to her brother. Her first cousin arrived and ate 2 sandwiches. Then her two other cousins arrived and ate 1 sandwich each. There were 3 sandwiches left. How many sandwiches did Ruth prepare?<|im_end|>
<|im_start|>assistant
<Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Ruth prepared sandwiches. She ate 1 sandwich and gave 2 sandwiches to her brother. Her first cousin arrived and ate 2 sandwiches. Then her two other cousins arrived and ate 1 sandwich each. There were 3 sandwiches left. How many sandwiches did Ruth prepare?<|im_end|>
<|im_start|>assistant
<Z_2><Z_41><Z_106><Z_109><Z_37><Z_52><Z_9><Z_77><Z_16><Z_81><Z_77><Z_103><ANSWER>

Question: Ruth prepared sandwiches. She ate 1 sandwich and gave 2 sandwiches to her brother. Her first cousin arrived and ate 2 sandwiches. Then her two other cousins arrived and ate 1 sandwich each. There were 3 sandwiches left. How many sandwiches did Ruth prepare?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6?<|im_end|>
<|im_start|>assistant
<Z_57><Z_115><Z_31><Z_32><Z_34><Z_120><Z_115><Z_66><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6?<|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
How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6?<|im_end|>
<|im_start|>assistant
<Z_78><Z_58><Z_45><Z_15><Z_66><Z_27><Z_120><Z_17><Z_53><ANSWER>

Question: How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paco had some sweet cookies and 11 salty cookies. He ate 5 sweet cookies and 2 salty cookies, and he ate 3 more sweet cookies than salty cookies. How many sweet cookies did he have initially?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paco had some sweet cookies and 11 salty cookies. He ate 5 sweet cookies and 2 salty cookies, and he ate 3 more sweet cookies than salty cookies. How many sweet cookies did he have initially?<|im_end|>
<|im_start|>assistant
<Z_104><Z_27><Z_58><Z_96><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paco had some sweet cookies and 11 salty cookies. He ate 5 sweet cookies and 2 salty cookies, and he ate 3 more sweet cookies than salty cookies. How many sweet cookies did he have initially?<|im_end|>
<|im_start|>assistant
<Z_102><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Paco had some sweet cookies and 11 salty cookies. He ate 5 sweet cookies and 2 salty cookies, and he ate 3 more sweet cookies than salty cookies. How many sweet cookies did he have initially?<|im_end|>
<|im_start|>assistant
<Z_91><Z_3><Z_8><Z_124><Z_50><ANSWER>

Question: Paco had some sweet cookies and 11 salty cookies. He ate 5 sweet cookies and 2 salty cookies, and he ate 3 more sweet cookies than salty cookies. How many sweet cookies did he have initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car rents for a certain amount per day plus 23 cents per mile. You are on a daily budget of 76 dollars and can go 200 miles. What is the daily rental cost of the car?<|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
A car rents for a certain amount per day plus 23 cents per mile. You are on a daily budget of 76 dollars and can go 200 miles. What is the daily rental cost of the car?<|im_end|>
<|im_start|>assistant
<Z_57><Z_108><Z_64><Z_58><Z_70><Z_58><Z_63><Z_58><Z_96><Z_52><Z_120><Z_76><Z_52><Z_96><Z_55><Z_16><Z_82><Z_95><Z_81><Z_55><Z_96><Z_126><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car rents for a certain amount per day plus 23 cents per mile. You are on a daily budget of 76 dollars and can go 200 miles. What is the daily rental cost of the car?<|im_end|>
<|im_start|>assistant
<Z_85><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A car rents for a certain amount per day plus 23 cents per mile. You are on a daily budget of 76 dollars and can go 200 miles. What is the daily rental cost of the car?<|im_end|>
<|im_start|>assistant
<Z_95><Z_108><Z_111><Z_120><Z_103><Z_32><Z_108><Z_67><Z_12><Z_89><Z_88><Z_32><Z_115><Z_52><Z_80><Z_56><Z_97><Z_123><Z_64><Z_85><Z_53><Z_123><Z_96><ANSWER>

Question: A car rents for a certain amount per day plus 23 cents per mile. You are on a daily budget of 76 dollars and can go 200 miles. What is the daily rental cost of the car?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?<|im_end|>
<|im_start|>assistant
<Z_113><Z_66><Z_99><Z_72><Z_62><Z_96><Z_71><Z_47><Z_39><Z_34><Z_123><Z_119><Z_17><Z_63><Z_92><Z_115><Z_120><Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?<|im_end|>
<|im_start|>assistant
<Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?<|im_end|>
<|im_start|>assistant
<Z_16><Z_85><Z_87><Z_39><Z_94><Z_92><Z_87><Z_37><Z_7><Z_53><Z_92><Z_30><Z_102><Z_110><Z_14><Z_29><Z_119><Z_83><ANSWER>

Question: Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $MN$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $MN$) are drawn through the rectangle?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $MN$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $MN$) are drawn through the rectangle?<|im_end|>
<|im_start|>assistant
<Z_113><Z_65><Z_72><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $MN$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $MN$) are drawn through the rectangle?<|im_end|>
<|im_start|>assistant
<Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the rectangle below, line segment $MN$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $MN$) are drawn through the rectangle?<|im_end|>
<|im_start|>assistant
<Z_100><Z_35><Z_87><Z_61><ANSWER>

Question: In the rectangle below, line segment $MN$ separates the rectangle into 2 sections. What is the largest number of sections into which the rectangle can be separated when $5$ line segments (including $MN$) are drawn through the rectangle?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest integer $b > 5$ for which the base $b$ number $43_b$ is a perfect cube?<|im_end|>
<|im_start|>assistant
<Z_104><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 integer $b > 5$ for which the base $b$ number $43_b$ is a perfect cube?<|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
What is the smallest integer $b > 5$ for which the base $b$ number $43_b$ is a perfect cube?<|im_end|>
<|im_start|>assistant
<Z_43><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 integer $b > 5$ for which the base $b$ number $43_b$ is a perfect cube?<|im_end|>
<|im_start|>assistant
<Z_32><ANSWER>

Question: What is the smallest integer $b > 5$ for which the base $b$ number $43_b$ is a perfect cube?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the focal distance of the hyperbola $2x^{2}-y^{2}=6$.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the focal distance of the hyperbola $2x^{2}-y^{2}=6$.<|im_end|>
<|im_start|>assistant
<Z_105><Z_103><Z_115><Z_63><Z_34><Z_34><Z_120><Z_95><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the focal distance of the hyperbola $2x^{2}-y^{2}=6$.<|im_end|>
<|im_start|>assistant
<Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the focal distance of the hyperbola $2x^{2}-y^{2}=6$.<|im_end|>
<|im_start|>assistant
<Z_50><Z_12><Z_115><Z_35><Z_42><Z_73><Z_40><Z_97><ANSWER>

Question: Find the focal distance of the hyperbola $2x^{2}-y^{2}=6$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The remainders when four positive integers are divided by 7 are 2, 3, 5, and 6. Find the remainder when the sum of the product of the first two integers and the product of the last two integers is divided by 7.<|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
The remainders when four positive integers are divided by 7 are 2, 3, 5, and 6. Find the remainder when the sum of the product of the first two integers and the product of the last two integers is divided by 7.<|im_end|>
<|im_start|>assistant
<Z_57><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The remainders when four positive integers are divided by 7 are 2, 3, 5, and 6. Find the remainder when the sum of the product of the first two integers and the product of the last two integers is divided by 7.<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The remainders when four positive integers are divided by 7 are 2, 3, 5, and 6. Find the remainder when the sum of the product of the first two integers and the product of the last two integers is divided by 7.<|im_end|>
<|im_start|>assistant
<Z_28><Z_20><ANSWER>

Question: The remainders when four positive integers are divided by 7 are 2, 3, 5, and 6. Find the remainder when the sum of the product of the first two integers and the product of the last two integers is divided by 7.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maria has a choice of 12 different muffins, 10 different kinds of butter, and 8 different kinds of jam. If she wants to make a breakfast set with one muffin, one kind of butter, and one kind of jam, how many different breakfast sets could she make?<|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
Maria has a choice of 12 different muffins, 10 different kinds of butter, and 8 different kinds of jam. If she wants to make a breakfast set with one muffin, one kind of butter, and one kind of jam, how many different breakfast sets could she make?<|im_end|>
<|im_start|>assistant
<Z_57><Z_66><Z_36><Z_37><Z_32><Z_88><Z_16><Z_53><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Maria has a choice of 12 different muffins, 10 different kinds of butter, and 8 different kinds of jam. If she wants to make a breakfast set with one muffin, one kind of butter, and one kind of jam, how many different breakfast sets could she make?<|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
Maria has a choice of 12 different muffins, 10 different kinds of butter, and 8 different kinds of jam. If she wants to make a breakfast set with one muffin, one kind of butter, and one kind of jam, how many different breakfast sets could she make?<|im_end|>
<|im_start|>assistant
<Z_16><Z_7><Z_31><Z_73><Z_58><Z_55><Z_114><Z_108><ANSWER>

Question: Maria has a choice of 12 different muffins, 10 different kinds of butter, and 8 different kinds of jam. If she wants to make a breakfast set with one muffin, one kind of butter, and one kind of jam, how many different breakfast sets could she make?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I need to subtract 23 from a certain number, but I mistakenly took 32 away, so I got 25. Find the correctly calculated value.<|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
I need to subtract 23 from a certain number, but I mistakenly took 32 away, so I got 25. Find the correctly calculated value.<|im_end|>
<|im_start|>assistant
<Z_94><Z_52><Z_40><Z_52><Z_96><Z_125><Z_80><Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I need to subtract 23 from a certain number, but I mistakenly took 32 away, so I got 25. Find the correctly calculated value.<|im_end|>
<|im_start|>assistant
<Z_13><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
I need to subtract 23 from a certain number, but I mistakenly took 32 away, so I got 25. Find the correctly calculated value.<|im_end|>
<|im_start|>assistant
<Z_31><Z_47><Z_61><Z_67><Z_50><Z_67><Z_37><Z_89><ANSWER>

Question: I need to subtract 23 from a certain number, but I mistakenly took 32 away, so I got 25. Find the correctly calculated value.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien has just reported that it has gathered $365_7$ units of a rare space mineral. Convert this amount from base 7 to base 10.<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien has just reported that it has gathered $365_7$ units of a rare space mineral. Convert this amount from base 7 to base 10.<|im_end|>
<|im_start|>assistant
<Z_94><Z_99><Z_125><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien has just reported that it has gathered $365_7$ units of a rare space mineral. Convert this amount from base 7 to base 10.<|im_end|>
<|im_start|>assistant
<Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
An alien has just reported that it has gathered $365_7$ units of a rare space mineral. Convert this amount from base 7 to base 10.<|im_end|>
<|im_start|>assistant
<Z_55><Z_4><Z_11><Z_8><ANSWER>

Question: An alien has just reported that it has gathered $365_7$ units of a rare space mineral. Convert this amount from base 7 to base 10.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If I have five boxes arranged in a row, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.)<|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
If I have five boxes arranged in a row, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.)<|im_end|>
<|im_start|>assistant
<Z_57><Z_125><Z_47><Z_54><Z_126><Z_123><Z_95><Z_61><Z_55><Z_91><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If I have five boxes arranged in a row, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.)<|im_end|>
<|im_start|>assistant
<Z_17><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If I have five boxes arranged in a row, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.)<|im_end|>
<|im_start|>assistant
<Z_33><Z_125><Z_0><Z_101><Z_126><Z_42><Z_38><Z_68><Z_73><Z_80><Z_68><ANSWER>

Question: If I have five boxes arranged in a row, in how many distinct ways can I place the digits $1$, $2$, $3$, and $4$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that the number of partitions of a set, denoted by \(S(n, k)\), satisfies the recurrence relation \(S(n+1, k) = S(n, k-1) + k \cdot S(n, k)\) for \(1 \leq k \leq n\). Use this relation to calculate \(S(5, 3)\).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that the number of partitions of a set, denoted by \(S(n, k)\), satisfies the recurrence relation \(S(n+1, k) = S(n, k-1) + k \cdot S(n, k)\) for \(1 \leq k \leq n\). Use this relation to calculate \(S(5, 3)\).<|im_end|>
<|im_start|>assistant
<Z_94><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that the number of partitions of a set, denoted by \(S(n, k)\), satisfies the recurrence relation \(S(n+1, k) = S(n, k-1) + k \cdot S(n, k)\) for \(1 \leq k \leq n\). Use this relation to calculate \(S(5, 3)\).<|im_end|>
<|im_start|>assistant
<Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that the number of partitions of a set, denoted by \(S(n, k)\), satisfies the recurrence relation \(S(n+1, k) = S(n, k-1) + k \cdot S(n, k)\) for \(1 \leq k \leq n\). Use this relation to calculate \(S(5, 3)\).<|im_end|>
<|im_start|>assistant
<Z_60><Z_57><ANSWER>

Question: Prove that the number of partitions of a set, denoted by \(S(n, k)\), satisfies the recurrence relation \(S(n+1, k) = S(n, k-1) + k \cdot S(n, k)\) for \(1 \leq k \leq n\). Use this relation to calculate \(S(5, 3)\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the number $2014ab$ is a multiple of both $\mathbf{3}$ and $\mathbf{5}$, how many such six-digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the number $2014ab$ is a multiple of both $\mathbf{3}$ and $\mathbf{5}$, how many such six-digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_94><Z_119><Z_119><Z_78><Z_75><Z_123><Z_103><Z_26><Z_10><Z_48><Z_114><Z_39><Z_101><Z_114><Z_10><Z_75><Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the number $2014ab$ is a multiple of both $\mathbf{3}$ and $\mathbf{5}$, how many such six-digit numbers are there?<|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
Given that the number $2014ab$ is a multiple of both $\mathbf{3}$ and $\mathbf{5}$, how many such six-digit numbers are there?<|im_end|>
<|im_start|>assistant
<Z_86><Z_40><Z_41><Z_57><Z_34><Z_40><Z_4><Z_18><Z_6><Z_125><Z_118><Z_107><Z_74><Z_47><Z_78><Z_98><Z_31><ANSWER>

Question: Given that the number $2014ab$ is a multiple of both $\mathbf{3}$ and $\mathbf{5}$, how many such six-digit numbers are there?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some positive real \(\alpha\), the set \(S\) of positive real numbers \(x\) with \(\{x\} > \alpha x\) consists of the union of several intervals, with total length 20.2. The value of \(\alpha\) can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\{x\} = x - \lfloor x \rfloor\) is the fractional part of \(x\).)<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some positive real \(\alpha\), the set \(S\) of positive real numbers \(x\) with \(\{x\} > \alpha x\) consists of the union of several intervals, with total length 20.2. The value of \(\alpha\) can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\{x\} = x - \lfloor x \rfloor\) is the fractional part of \(x\).)<|im_end|>
<|im_start|>assistant
<Z_104><Z_62><Z_63><Z_64><Z_93><Z_87><Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some positive real \(\alpha\), the set \(S\) of positive real numbers \(x\) with \(\{x\} > \alpha x\) consists of the union of several intervals, with total length 20.2. The value of \(\alpha\) can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\{x\} = x - \lfloor x \rfloor\) is the fractional part of \(x\).)<|im_end|>
<|im_start|>assistant
<Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For some positive real \(\alpha\), the set \(S\) of positive real numbers \(x\) with \(\{x\} > \alpha x\) consists of the union of several intervals, with total length 20.2. The value of \(\alpha\) can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\{x\} = x - \lfloor x \rfloor\) is the fractional part of \(x\).)<|im_end|>
<|im_start|>assistant
<Z_5><Z_121><Z_47><Z_105><Z_97><Z_15><Z_101><ANSWER>

Question: For some positive real \(\alpha\), the set \(S\) of positive real numbers \(x\) with \(\{x\} > \alpha x\) consists of the union of several intervals, with total length 20.2. The value of \(\alpha\) can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\). (Here, \(\{x\} = x - \lfloor x \rfloor\) is the fractional part of \(x\).)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covered x km at 50 kmph and another 2x km at a certain speed. The average speed of the train in covering the entire 3x km is 25 kmph. What is the speed of the train during the 2x km?<|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
A train covered x km at 50 kmph and another 2x km at a certain speed. The average speed of the train in covering the entire 3x km is 25 kmph. What is the speed of the train during the 2x km?<|im_end|>
<|im_start|>assistant
<Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train covered x km at 50 kmph and another 2x km at a certain speed. The average speed of the train in covering the entire 3x km is 25 kmph. What is the speed of the train during the 2x km?<|im_end|>
<|im_start|>assistant
<Z_41><ANSWER>

Question: A train covered x km at 50 kmph and another 2x km at a certain speed. The average speed of the train in covering the entire 3x km is 25 kmph. What is the speed of the train during the 2x km?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the school bake sale Robin made 42 cupcakes. She sold some of them and then made 39 more. She ended up with 59 cupcakes. How many cupcakes did she sell?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the school bake sale Robin made 42 cupcakes. She sold some of them and then made 39 more. She ended up with 59 cupcakes. How many cupcakes did she sell?<|im_end|>
<|im_start|>assistant
<Z_104><Z_34><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the school bake sale Robin made 42 cupcakes. She sold some of them and then made 39 more. She ended up with 59 cupcakes. How many cupcakes did she sell?<|im_end|>
<|im_start|>assistant
<Z_52><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For the school bake sale Robin made 42 cupcakes. She sold some of them and then made 39 more. She ended up with 59 cupcakes. How many cupcakes did she sell?<|im_end|>
<|im_start|>assistant
<Z_92><Z_26><Z_65><ANSWER>

Question: For the school bake sale Robin made 42 cupcakes. She sold some of them and then made 39 more. She ended up with 59 cupcakes. How many cupcakes did she sell?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many of the first $2000$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many of the first $2000$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_89><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many of the first $2000$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?<|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many of the first $2000$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?<|im_end|>
<|im_start|>assistant
<Z_11><Z_62><ANSWER>

Question:
How many of the first $2000$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many divisors of 150 are also divisors of 45?<|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
How many divisors of 150 are also divisors of 45?<|im_end|>
<|im_start|>assistant
<Z_57><Z_108><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many divisors of 150 are also divisors of 45?<|im_end|>
<|im_start|>assistant
<Z_76><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many divisors of 150 are also divisors of 45?<|im_end|>
<|im_start|>assistant
<Z_65><Z_119><Z_29><ANSWER>

Question: How many divisors of 150 are also divisors of 45?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 7 classrooms. There are 68 boys and 53 girls. How many total students can be in each classroom while maintaining an equal distribution of boys and girls in each room?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 7 classrooms. There are 68 boys and 53 girls. How many total students can be in each classroom while maintaining an equal distribution of boys and girls in each room?<|im_end|>
<|im_start|>assistant
<Z_94><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 7 classrooms. There are 68 boys and 53 girls. How many total students can be in each classroom while maintaining an equal distribution of boys and girls in each room?<|im_end|>
<|im_start|>assistant
<Z_44><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 7 classrooms. There are 68 boys and 53 girls. How many total students can be in each classroom while maintaining an equal distribution of boys and girls in each room?<|im_end|>
<|im_start|>assistant
<Z_5><Z_81><ANSWER>

Question: The school principal decided that she wanted every class to have an equal number of boys and girls in each first-grade classroom. There are 7 classrooms. There are 68 boys and 53 girls. How many total students can be in each classroom while maintaining an equal distribution of boys and girls in each room?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a, b, c, d\) be positive integers such that
\[
\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 15 & 10 \\ -9 & -6 \end{pmatrix}.
\]
Find the smallest possible value of \(a + b + c + d\).<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a, b, c, d\) be positive integers such that
\[
\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 15 & 10 \\ -9 & -6 \end{pmatrix}.
\]
Find the smallest possible value of \(a + b + c + d\).<|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
Let \(a, b, c, d\) be positive integers such that
\[
\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 15 & 10 \\ -9 & -6 \end{pmatrix}.
\]
Find the smallest possible value of \(a + b + c + d\).<|im_end|>
<|im_start|>assistant
<Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \(a, b, c, d\) be positive integers such that
\[
\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 15 & 10 \\ -9 & -6 \end{pmatrix}.
\]
Find the smallest possible value of \(a + b + c + d\).<|im_end|>
<|im_start|>assistant
<Z_91><ANSWER>

Question:
Let \(a, b, c, d\) be positive integers such that
\[
\begin{pmatrix} 5 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 15 & 10 \\ -9 & -6 \end{pmatrix}.
\]
Find the smallest possible value of \(a + b + c + d\).


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tommy is making steaks for his family. There are some members in his family. If each member wants one pound and the steaks are 20 ounces each, Tommy needs to buy 4 steaks. How many members are in Tommy's family?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tommy is making steaks for his family. There are some members in his family. If each member wants one pound and the steaks are 20 ounces each, Tommy needs to buy 4 steaks. How many members are in Tommy's family?<|im_end|>
<|im_start|>assistant
<Z_104><Z_86><Z_59><Z_108><Z_93><Z_11><Z_34><Z_114><Z_91><Z_126><Z_83><Z_82><Z_19><Z_93><Z_39><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tommy is making steaks for his family. There are some members in his family. If each member wants one pound and the steaks are 20 ounces each, Tommy needs to buy 4 steaks. How many members are in Tommy's family?<|im_end|>
<|im_start|>assistant
<Z_127><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tommy is making steaks for his family. There are some members in his family. If each member wants one pound and the steaks are 20 ounces each, Tommy needs to buy 4 steaks. How many members are in Tommy's family?<|im_end|>
<|im_start|>assistant
<Z_100><Z_37><Z_66><Z_104><Z_46><Z_24><Z_1><Z_2><Z_49><Z_9><Z_42><Z_3><Z_78><Z_42><Z_63><ANSWER>

Question: Tommy is making steaks for his family. There are some members in his family. If each member wants one pound and the steaks are 20 ounces each, Tommy needs to buy 4 steaks. How many members are in Tommy's family?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{2008}x_n$.
<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{2008}x_n$.
<|im_end|>
<|im_start|>assistant
<Z_104><Z_27><Z_58><Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{2008}x_n$.
<|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{2008}x_n$.
<|im_end|>
<|im_start|>assistant
<Z_72><Z_59><Z_52><Z_15><ANSWER>

Question: Suppose that $x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{2008}x_n$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define the function $g$ on positive integers:
\[ g(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
g(n - 6) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function $g$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define the function $g$ on positive integers:
\[ g(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
g(n - 6) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function $g$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_97><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define the function $g$ on positive integers:
\[ g(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
g(n - 6) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function $g$.<|im_end|>
<|im_start|>assistant
<Z_106><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Define the function $g$ on positive integers:
\[ g(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
g(n - 6) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function $g$.<|im_end|>
<|im_start|>assistant
<Z_11><Z_92><Z_93><ANSWER>

Question:
Define the function $g$ on positive integers:
\[ g(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
g(n - 6) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function $g$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?<|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
10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?<|im_end|>
<|im_start|>assistant
<Z_94><Z_37><Z_100><Z_100><Z_100><Z_126><Z_54><Z_32><Z_92><Z_10><Z_119><Z_93><Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?<|im_end|>
<|im_start|>assistant
<Z_0><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?<|im_end|>
<|im_start|>assistant
<Z_95><Z_51><Z_117><Z_112><Z_63><Z_74><Z_88><Z_40><Z_77><Z_104><Z_120><Z_107><Z_109><ANSWER>

Question: 10 runners start at the same time: five in blue jerseys from one end of the running track, and five in red jerseys from the other. Their speeds are constant and different, with each runner’s speed being more than 9 km/h but less than 12 km/h. Upon reaching the end of the track, each runner immediately runs back, and upon returning to their starting point, they finish the run. The coach marks a check each time two runners in different colored jerseys meet (either face to face or one catching up with the other) (more than two runners do not meet at a point during the run). How many checks will the coach make by the time the fastest runner finishes their run?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

a) Write down all the permutations of the elements 1, 2, 3, 4 in this unusual arrangement!

b) What is the rank of the permutation 5341726 of the elements 1, 2, 3, 4, 5, 6, 7

I. in this unusual arrangement?

II. in the usual arrangement?<|im_end|>
<|im_start|>assistant
<Z_94><Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

a) Write down all the permutations of the elements 1, 2, 3, 4 in this unusual arrangement!

b) What is the rank of the permutation 5341726 of the elements 1, 2, 3, 4, 5, 6, 7

I. in this unusual arrangement?

II. in the usual arrangement?<|im_end|>
<|im_start|>assistant
<Z_57><Z_29><Z_100><Z_68><Z_50><Z_34><Z_124><Z_49><Z_92><Z_11><Z_88><Z_72><Z_82><Z_31><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

a) Write down all the permutations of the elements 1, 2, 3, 4 in this unusual arrangement!

b) What is the rank of the permutation 5341726 of the elements 1, 2, 3, 4, 5, 6, 7

I. in this unusual arrangement?

II. in the usual arrangement?<|im_end|>
<|im_start|>assistant
<Z_89><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user

a) Write down all the permutations of the elements 1, 2, 3, 4 in this unusual arrangement!

b) What is the rank of the permutation 5341726 of the elements 1, 2, 3, 4, 5, 6, 7

I. in this unusual arrangement?

II. in the usual arrangement?<|im_end|>
<|im_start|>assistant
<Z_108><Z_90><Z_105><Z_64><Z_104><Z_83><Z_38><Z_73><Z_13><Z_37><Z_109><Z_82><Z_85><Z_53><Z_19><ANSWER>

Question:
a) Write down all the permutations of the elements 1, 2, 3, 4 in this unusual arrangement!

b) What is the rank of the permutation 5341726 of the elements 1, 2, 3, 4, 5, 6, 7

I. in this unusual arrangement?

II. in the usual arrangement?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan is creating a playlist for his 90-minute run. He has 10 songs that are 3 minutes long, 12 songs that are 4 minutes long, and 15 songs that are 6 minutes long. Stan can only include up to 7 songs from each category in his playlist. Given these limitations, he wants to make sure his playlist also contains at least 3 songs from his favorite band. Each song from his favorite band is 4 minutes long. How many more minutes of songs will he need in his playlist if he wants it to cover the entire run and fulfill the desired conditions?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan is creating a playlist for his 90-minute run. He has 10 songs that are 3 minutes long, 12 songs that are 4 minutes long, and 15 songs that are 6 minutes long. Stan can only include up to 7 songs from each category in his playlist. Given these limitations, he wants to make sure his playlist also contains at least 3 songs from his favorite band. Each song from his favorite band is 4 minutes long. How many more minutes of songs will he need in his playlist if he wants it to cover the entire run and fulfill the desired conditions?<|im_end|>
<|im_start|>assistant
<Z_57><Z_99><Z_107><Z_124><Z_49><Z_119><Z_12><Z_56><Z_115><Z_65><Z_125><Z_12><Z_47><Z_31><Z_37><Z_36><Z_103><Z_73><Z_47><Z_53><Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan is creating a playlist for his 90-minute run. He has 10 songs that are 3 minutes long, 12 songs that are 4 minutes long, and 15 songs that are 6 minutes long. Stan can only include up to 7 songs from each category in his playlist. Given these limitations, he wants to make sure his playlist also contains at least 3 songs from his favorite band. Each song from his favorite band is 4 minutes long. How many more minutes of songs will he need in his playlist if he wants it to cover the entire run and fulfill the desired conditions?<|im_end|>
<|im_start|>assistant
<Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Stan is creating a playlist for his 90-minute run. He has 10 songs that are 3 minutes long, 12 songs that are 4 minutes long, and 15 songs that are 6 minutes long. Stan can only include up to 7 songs from each category in his playlist. Given these limitations, he wants to make sure his playlist also contains at least 3 songs from his favorite band. Each song from his favorite band is 4 minutes long. How many more minutes of songs will he need in his playlist if he wants it to cover the entire run and fulfill the desired conditions?<|im_end|>
<|im_start|>assistant
<Z_57><Z_91><Z_95><Z_28><Z_49><Z_82><Z_87><Z_45><Z_46><Z_5><Z_111><Z_102><Z_105><Z_33><Z_123><Z_73><Z_62><Z_117><Z_78><Z_97><Z_112><ANSWER>

Question: Stan is creating a playlist for his 90-minute run. He has 10 songs that are 3 minutes long, 12 songs that are 4 minutes long, and 15 songs that are 6 minutes long. Stan can only include up to 7 songs from each category in his playlist. Given these limitations, he wants to make sure his playlist also contains at least 3 songs from his favorite band. Each song from his favorite band is 4 minutes long. How many more minutes of songs will he need in his playlist if he wants it to cover the entire run and fulfill the desired conditions?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_68><Z_81><Z_80><Z_12><Z_96><Z_125><Z_70><Z_115><Z_20><Z_98><Z_47><Z_45><Z_91><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.<|im_end|>
<|im_start|>assistant
<Z_59><Z_89><Z_100><Z_109><Z_11><Z_78><Z_106><Z_38><Z_80><Z_31><Z_95><Z_75><Z_69><Z_100><Z_47><ANSWER>

Question: Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Baker made 149 cakes and some pastries. He sold 10 cakes and 90 pastries. Baker still has 139 cakes. How many pastries did he make initially?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Baker made 149 cakes and some pastries. He sold 10 cakes and 90 pastries. Baker still has 139 cakes. How many pastries did he make initially?<|im_end|>
<|im_start|>assistant
<Z_104><Z_27><Z_115><Z_40><Z_125><Z_99><Z_56><Z_125><Z_41><Z_87><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Baker made 149 cakes and some pastries. He sold 10 cakes and 90 pastries. Baker still has 139 cakes. How many pastries did he make initially?<|im_end|>
<|im_start|>assistant
<Z_11><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Baker made 149 cakes and some pastries. He sold 10 cakes and 90 pastries. Baker still has 139 cakes. How many pastries did he make initially?<|im_end|>
<|im_start|>assistant
<Z_91><Z_18><Z_51><Z_96><Z_102><Z_64><Z_78><Z_14><Z_23><Z_127><Z_7><ANSWER>

Question: Baker made 149 cakes and some pastries. He sold 10 cakes and 90 pastries. Baker still has 139 cakes. How many pastries did he make initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 3)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 3)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_27><Z_38><Z_70><Z_88><Z_71><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 3)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_68><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 3)^{x^2 - 7x + 6} = 1.\]<|im_end|>
<|im_start|>assistant
<Z_101><Z_68><Z_114><Z_20><Z_72><ANSWER>

Question:
Determine the sum of all real numbers \(x\) satisfying
\[(x^2 - 6x + 3)^{x^2 - 7x + 6} = 1.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 19x + 88$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 23x + 132$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 19x + 88$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 23x + 132$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_74><Z_43><Z_1><Z_91><Z_37><Z_63><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 19x + 88$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 23x + 132$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?<|im_end|>
<|im_start|>assistant
<Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The expression $x^2 + 19x + 88$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 23x + 132$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?<|im_end|>
<|im_start|>assistant
<Z_4><Z_36><Z_74><Z_77><Z_11><Z_64><Z_82><ANSWER>

Question: The expression $x^2 + 19x + 88$ can be written as $(x + a)(x + b)$, and the expression $x^2 - 23x + 132$ can be written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is 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
if one person completes a journey in 10 hrs . he travels 1 st half of the journey at the rate of 21 km / hr and 2 nd half at therate of 24 km / hr . find the total journey in km .<|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
if one person completes a journey in 10 hrs . he travels 1 st half of the journey at the rate of 21 km / hr and 2 nd half at therate of 24 km / hr . find the total journey in km .<|im_end|>
<|im_start|>assistant
<Z_57><Z_62><Z_80><Z_120><Z_107><Z_34><Z_82><Z_79><Z_79><Z_55><Z_120><Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if one person completes a journey in 10 hrs . he travels 1 st half of the journey at the rate of 21 km / hr and 2 nd half at therate of 24 km / hr . find the total journey in km .<|im_end|>
<|im_start|>assistant
<Z_56><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
if one person completes a journey in 10 hrs . he travels 1 st half of the journey at the rate of 21 km / hr and 2 nd half at therate of 24 km / hr . find the total journey in km .<|im_end|>
<|im_start|>assistant
<Z_50><Z_29><Z_92><Z_49><Z_113><Z_48><Z_58><Z_114><Z_66><Z_2><Z_44><Z_116><ANSWER>

Question: if one person completes a journey in 10 hrs . he travels 1 st half of the journey at the rate of 21 km / hr and 2 nd half at therate of 24 km / hr . find the total journey in km .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.<|im_end|>
<|im_start|>assistant
<Z_78><ANSWER>

Question: Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A magician was selling magic card decks for 7 dollars each. If he started with 16 decks and by the end of the day he had 8 left, how much money did he earn? <|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
 A magician was selling magic card decks for 7 dollars each. If he started with 16 decks and by the end of the day he had 8 left, how much money did he earn? <|im_end|>
<|im_start|>assistant
<Z_113><Z_108><Z_74><Z_38><Z_40><Z_58><Z_107><Z_103><Z_73><Z_59><Z_70><Z_76><Z_82><Z_34><Z_12><Z_80><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A magician was selling magic card decks for 7 dollars each. If he started with 16 decks and by the end of the day he had 8 left, how much money did he earn? <|im_end|>
<|im_start|>assistant
<Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
 A magician was selling magic card decks for 7 dollars each. If he started with 16 decks and by the end of the day he had 8 left, how much money did he earn? <|im_end|>
<|im_start|>assistant
<Z_57><Z_62><Z_37><Z_119><Z_90><Z_2><Z_52><Z_36><Z_86><Z_38><Z_42><Z_22><Z_2><Z_112><Z_43><Z_5><ANSWER>

Question: A magician was selling magic card decks for 7 dollars each. If he started with 16 decks and by the end of the day he had 8 left, how much money did he earn?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 34 coloring books. She gave away some of them, but then bought 48 more. Now, she has a total of 79 coloring books. How many coloring books did Faye give away?<|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
Faye had 34 coloring books. She gave away some of them, but then bought 48 more. Now, she has a total of 79 coloring books. How many coloring books did Faye give away?<|im_end|>
<|im_start|>assistant
<Z_104><Z_72><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 34 coloring books. She gave away some of them, but then bought 48 more. Now, she has a total of 79 coloring books. How many coloring books did Faye give away?<|im_end|>
<|im_start|>assistant
<Z_70><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Faye had 34 coloring books. She gave away some of them, but then bought 48 more. Now, she has a total of 79 coloring books. How many coloring books did Faye give away?<|im_end|>
<|im_start|>assistant
<Z_89><Z_6><Z_61><ANSWER>

Question: Faye had 34 coloring books. She gave away some of them, but then bought 48 more. Now, she has a total of 79 coloring books. How many coloring books did Faye give away?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adding 8 consecutive even numbers gives 424. Find the last number out of 8 even numbers.<|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
Adding 8 consecutive even numbers gives 424. Find the last number out of 8 even numbers.<|im_end|>
<|im_start|>assistant
<Z_113><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adding 8 consecutive even numbers gives 424. Find the last number out of 8 even numbers.<|im_end|>
<|im_start|>assistant
<Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Adding 8 consecutive even numbers gives 424. Find the last number out of 8 even numbers.<|im_end|>
<|im_start|>assistant
<Z_110><ANSWER>

Question: Adding 8 consecutive even numbers gives 424. Find the last number out of 8 even numbers.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a work in 3 days . b can do the same work in 6 days . if both a & b are working together in how many days they will finish the work ?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a work in 3 days . b can do the same work in 6 days . if both a & b are working together in how many days they will finish the work ?<|im_end|>
<|im_start|>assistant
<Z_104><Z_79><Z_91><Z_99><Z_34><Z_82><Z_59><Z_61><Z_61><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a work in 3 days . b can do the same work in 6 days . if both a & b are working together in how many days they will finish the work ?<|im_end|>
<|im_start|>assistant
<Z_69><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
a can do a work in 3 days . b can do the same work in 6 days . if both a & b are working together in how many days they will finish the work ?<|im_end|>
<|im_start|>assistant
<Z_103><Z_121><Z_113><Z_75><Z_121><Z_63><Z_36><Z_50><Z_93><ANSWER>

Question: a can do a work in 3 days . b can do the same work in 6 days . if both a & b are working together in how many days they will finish the work ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The king decided to test his one hundred wise men by conducting the following experiment: he will line them up with their eyes blindfolded, then place a black or white cap on each of their heads. After the blindfolds are removed, each wise man, starting from the last in the line, must state the color of his cap. If he guesses incorrectly, he will be executed. The wise men have some time to strategize before the test. How many wise men are guaranteed to be saved?<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The king decided to test his one hundred wise men by conducting the following experiment: he will line them up with their eyes blindfolded, then place a black or white cap on each of their heads. After the blindfolds are removed, each wise man, starting from the last in the line, must state the color of his cap. If he guesses incorrectly, he will be executed. The wise men have some time to strategize before the test. How many wise men are guaranteed to be saved?<|im_end|>
<|im_start|>assistant
<Z_113><Z_73><Z_73><Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The king decided to test his one hundred wise men by conducting the following experiment: he will line them up with their eyes blindfolded, then place a black or white cap on each of their heads. After the blindfolds are removed, each wise man, starting from the last in the line, must state the color of his cap. If he guesses incorrectly, he will be executed. The wise men have some time to strategize before the test. How many wise men are guaranteed to be saved?<|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
The king decided to test his one hundred wise men by conducting the following experiment: he will line them up with their eyes blindfolded, then place a black or white cap on each of their heads. After the blindfolds are removed, each wise man, starting from the last in the line, must state the color of his cap. If he guesses incorrectly, he will be executed. The wise men have some time to strategize before the test. How many wise men are guaranteed to be saved?<|im_end|>
<|im_start|>assistant
<Z_50><Z_67><Z_55><Z_78><ANSWER>

Question: The king decided to test his one hundred wise men by conducting the following experiment: he will line them up with their eyes blindfolded, then place a black or white cap on each of their heads. After the blindfolds are removed, each wise man, starting from the last in the line, must state the color of his cap. If he guesses incorrectly, he will be executed. The wise men have some time to strategize before the test. How many wise men are guaranteed to be saved?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:<|im_end|>
<|im_start|>assistant
<Z_104><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:<|im_end|>
<|im_start|>assistant
<Z_104><Z_51><Z_107><Z_99><Z_107><Z_87><Z_91><Z_20><Z_117><Z_34><Z_20><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:<|im_end|>
<|im_start|>assistant
<Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:<|im_end|>
<|im_start|>assistant
<Z_84><Z_50><Z_62><Z_97><Z_7><Z_127><Z_80><Z_69><Z_57><Z_92><Z_29><ANSWER>

Question: There are 10 mountaineers, divided equally into two groups. Among them, 4 are familiar with the trails. Each group needs 2 people who are familiar with the trails. The number of different ways to distribute them is:


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A woman goes to work and has an average speed of 45 mph . She takes the same route home and has an average speed of 30 mph . Her commute to and from work took an hour . How many miles does she travel to work ?<|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
A woman goes to work and has an average speed of 45 mph . She takes the same route home and has an average speed of 30 mph . Her commute to and from work took an hour . How many miles does she travel to work ?<|im_end|>
<|im_start|>assistant
<Z_57><Z_89><Z_78><Z_73><Z_107><Z_36><Z_125><Z_28><Z_121><Z_48><Z_28><Z_80><Z_70><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A woman goes to work and has an average speed of 45 mph . She takes the same route home and has an average speed of 30 mph . Her commute to and from work took an hour . How many miles does she travel to work ?<|im_end|>
<|im_start|>assistant
<Z_86><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A woman goes to work and has an average speed of 45 mph . She takes the same route home and has an average speed of 30 mph . Her commute to and from work took an hour . How many miles does she travel to work ?<|im_end|>
<|im_start|>assistant
<Z_25><Z_111><Z_20><Z_93><Z_27><Z_9><Z_46><Z_4><Z_110><Z_54><Z_52><Z_14><Z_6><ANSWER>

Question: A woman goes to work and has an average speed of 45 mph . She takes the same route home and has an average speed of 30 mph . Her commute to and from work took an hour . How many miles does she travel to work ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mom preserves plums in jars so that the plums from one jar are enough for either 16 turnovers, or 4 cakes, or half a tray of fruit bars.

She has 4 such jars in her pantry and wants to bake one tray of fruit bars and 6 cakes. How many turnovers can she make with the remaining plums?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mom preserves plums in jars so that the plums from one jar are enough for either 16 turnovers, or 4 cakes, or half a tray of fruit bars.

She has 4 such jars in her pantry and wants to bake one tray of fruit bars and 6 cakes. How many turnovers can she make with the remaining plums?<|im_end|>
<|im_start|>assistant
<Z_104><Z_37><Z_82><Z_87><Z_114><Z_34><Z_92><Z_98><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mom preserves plums in jars so that the plums from one jar are enough for either 16 turnovers, or 4 cakes, or half a tray of fruit bars.

She has 4 such jars in her pantry and wants to bake one tray of fruit bars and 6 cakes. How many turnovers can she make with the remaining plums?<|im_end|>
<|im_start|>assistant
<Z_120><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mom preserves plums in jars so that the plums from one jar are enough for either 16 turnovers, or 4 cakes, or half a tray of fruit bars.

She has 4 such jars in her pantry and wants to bake one tray of fruit bars and 6 cakes. How many turnovers can she make with the remaining plums?<|im_end|>
<|im_start|>assistant
<Z_28><Z_34><Z_89><Z_100><Z_46><Z_97><Z_3><Z_12><ANSWER>

Question:
Mom preserves plums in jars so that the plums from one jar are enough for either 16 turnovers, or 4 cakes, or half a tray of fruit bars.

She has 4 such jars in her pantry and wants to bake one tray of fruit bars and 6 cakes. How many turnovers can she make with the remaining plums?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can five students stand in a straight line if two specific students refuse to stand next to each other?<|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
In how many different ways can five students stand in a straight line if two specific students refuse to stand next to each other?<|im_end|>
<|im_start|>assistant
<Z_113><Z_70><Z_59><Z_93><Z_70><Z_79><Z_117><Z_54><Z_93><Z_16><Z_82><Z_93><Z_63><Z_54><Z_117><Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can five students stand in a straight line if two specific students refuse to stand next to each other?<|im_end|>
<|im_start|>assistant
<Z_78><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many different ways can five students stand in a straight line if two specific students refuse to stand next to each other?<|im_end|>
<|im_start|>assistant
<Z_45><Z_34><Z_41><Z_56><Z_55><Z_67><Z_84><Z_117><Z_109><Z_74><Z_12><Z_9><Z_48><Z_21><Z_13><Z_64><ANSWER>

Question: In how many different ways can five students stand in a straight line if two specific students refuse to stand next to each other?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of 11 numbers is 60. Out of 11 numbers, the average of the first 6 numbers is 98, and the average of the last 6 numbers is 65. What is the value of the 6th number?<|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
The average of 11 numbers is 60. Out of 11 numbers, the average of the first 6 numbers is 98, and the average of the last 6 numbers is 65. What is the value of the 6th number?<|im_end|>
<|im_start|>assistant
<Z_104><Z_100><Z_33><Z_73><Z_120><Z_99><Z_45><Z_10><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of 11 numbers is 60. Out of 11 numbers, the average of the first 6 numbers is 98, and the average of the last 6 numbers is 65. What is the value of the 6th number?<|im_end|>
<|im_start|>assistant
<Z_85><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The average of 11 numbers is 60. Out of 11 numbers, the average of the first 6 numbers is 98, and the average of the last 6 numbers is 65. What is the value of the 6th number?<|im_end|>
<|im_start|>assistant
<Z_18><Z_54><Z_104><Z_13><Z_53><Z_19><Z_115><Z_105><Z_101><ANSWER>

Question: The average of 11 numbers is 60. Out of 11 numbers, the average of the first 6 numbers is 98, and the average of the last 6 numbers is 65. What is the value of the 6th number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carolyn starts with 47 marbles and 6 oranges. She shares some marbles with Diana and ends up with 5 marbles. How many marbles did Carolyn share with Diana?<|im_end|>
<|im_start|>assistant
<Z_57><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carolyn starts with 47 marbles and 6 oranges. She shares some marbles with Diana and ends up with 5 marbles. How many marbles did Carolyn share with Diana?<|im_end|>
<|im_start|>assistant
<Z_113><Z_77><Z_47><Z_103><Z_66><Z_96><Z_121><Z_114><Z_126><Z_117><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carolyn starts with 47 marbles and 6 oranges. She shares some marbles with Diana and ends up with 5 marbles. How many marbles did Carolyn share with Diana?<|im_end|>
<|im_start|>assistant
<Z_18><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Carolyn starts with 47 marbles and 6 oranges. She shares some marbles with Diana and ends up with 5 marbles. How many marbles did Carolyn share with Diana?<|im_end|>
<|im_start|>assistant
<Z_66><Z_81><Z_68><Z_93><Z_114><Z_54><Z_111><Z_70><Z_80><Z_114><ANSWER>

Question: Carolyn starts with 47 marbles and 6 oranges. She shares some marbles with Diana and ends up with 5 marbles. How many marbles did Carolyn share with Diana?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cistern is normally filled in 8 hours but takes two hours longer to fill because of three leaks in its bottom. Leak A can empty the full cistern in "x" hours, Leak B in "y" hours, and Leak C in "z" hours. If the cistern is full, in how many hours will all three leaks together completely empty it?<|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
A cistern is normally filled in 8 hours but takes two hours longer to fill because of three leaks in its bottom. Leak A can empty the full cistern in "x" hours, Leak B in "y" hours, and Leak C in "z" hours. If the cistern is full, in how many hours will all three leaks together completely empty it?<|im_end|>
<|im_start|>assistant
<Z_57><Z_119><Z_40><Z_54><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cistern is normally filled in 8 hours but takes two hours longer to fill because of three leaks in its bottom. Leak A can empty the full cistern in "x" hours, Leak B in "y" hours, and Leak C in "z" hours. If the cistern is full, in how many hours will all three leaks together completely empty it?<|im_end|>
<|im_start|>assistant
<Z_67><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cistern is normally filled in 8 hours but takes two hours longer to fill because of three leaks in its bottom. Leak A can empty the full cistern in "x" hours, Leak B in "y" hours, and Leak C in "z" hours. If the cistern is full, in how many hours will all three leaks together completely empty it?<|im_end|>
<|im_start|>assistant
<Z_66><Z_46><Z_105><Z_72><ANSWER>

Question: A cistern is normally filled in 8 hours but takes two hours longer to fill because of three leaks in its bottom. Leak A can empty the full cistern in "x" hours, Leak B in "y" hours, and Leak C in "z" hours. If the cistern is full, in how many hours will all three leaks together completely empty it?