<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|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
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|im_end|>
<|im_start|>assistant
<Z_94><Z_74><Z_49><Z_58><Z_49><Z_107><Z_66><Z_82><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|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
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)<|im_end|>
<|im_start|>assistant
<Z_77><Z_81><Z_103><Z_13><Z_127><Z_48><Z_25><Z_102><ANSWER>

Question: Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna 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
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|im_end|>
<|im_start|>assistant
<Z_113><Z_119><Z_114><Z_65><Z_50><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|im_end|>
<|im_start|>assistant
<Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?<|im_end|>
<|im_start|>assistant
<Z_114><Z_0><Z_34><Z_1><Z_69><ANSWER>

Question: Joanna and Jacques had some gumballs in their dishes. Jacques had 60 gumballs. They then purchased 4 times the number of gumballs they had initially and added these to their dishes. When they decided to put together their gumballs and shared them equally, each got 250 gumballs. How many gumballs did Joanna have initially?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|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
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_29><Z_47><Z_59><Z_72><Z_80><Z_31><Z_76><Z_45><Z_61><Z_90><Z_58><Z_11><Z_33><Z_75><Z_91><Z_80><Z_115><Z_55><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_123><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?<|im_end|>
<|im_start|>assistant
<Z_0><Z_33><Z_56><Z_59><Z_79><Z_43><Z_105><Z_91><Z_73><Z_40><Z_8><Z_66><Z_127><Z_9><Z_31><Z_57><Z_48><Z_85><ANSWER>

Question: Tickets for the homecoming dance cost $20 for a single ticket or $35 for a couple. Ticket sales totaled $2280, and 128 people attended. How many couple tickets were sold?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain sports equipment store is preparing to purchase two types of table tennis balls, brand A and brand B. If they buy 10 balls of brand A and 5 balls of brand B, it will cost $100. If they buy 5 balls of brand A and 3 balls of brand B, it will cost $55.
$(1)$ Find the cost of each ball of brand A and brand B.
$(2)$ If the sports equipment store spends exactly $1000 to purchase these two types of table tennis balls, considering customer demand, they require the quantity of brand A balls to be at least 6 times the quantity of brand B balls, and the quantity of brand B balls to be at least 23. How many different purchasing plans does the store have?
$(3)$ If the store makes a profit of $3 for each ball of brand A sold and $4 for each ball of brand B sold, which purchasing plan from question $(2)$ yields the highest profit, and what is the maximum profit?<|im_end|>
<|im_start|>assistant
<Z_57><Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain sports equipment store is preparing to purchase two types of table tennis balls, brand A and brand B. If they buy 10 balls of brand A and 5 balls of brand B, it will cost $100. If they buy 5 balls of brand A and 3 balls of brand B, it will cost $55.
$(1)$ Find the cost of each ball of brand A and brand B.
$(2)$ If the sports equipment store spends exactly $1000 to purchase these two types of table tennis balls, considering customer demand, they require the quantity of brand A balls to be at least 6 times the quantity of brand B balls, and the quantity of brand B balls to be at least 23. How many different purchasing plans does the store have?
$(3)$ If the store makes a profit of $3 for each ball of brand A sold and $4 for each ball of brand B sold, which purchasing plan from question $(2)$ yields the highest profit, and what is the maximum profit?<|im_end|>
<|im_start|>assistant
<Z_94><Z_49><Z_65><Z_51><Z_89><Z_117><Z_68><Z_101><Z_87><Z_79><Z_28><Z_48><Z_126><Z_107><Z_32><Z_38><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain sports equipment store is preparing to purchase two types of table tennis balls, brand A and brand B. If they buy 10 balls of brand A and 5 balls of brand B, it will cost $100. If they buy 5 balls of brand A and 3 balls of brand B, it will cost $55.
$(1)$ Find the cost of each ball of brand A and brand B.
$(2)$ If the sports equipment store spends exactly $1000 to purchase these two types of table tennis balls, considering customer demand, they require the quantity of brand A balls to be at least 6 times the quantity of brand B balls, and the quantity of brand B balls to be at least 23. How many different purchasing plans does the store have?
$(3)$ If the store makes a profit of $3 for each ball of brand A sold and $4 for each ball of brand B sold, which purchasing plan from question $(2)$ yields the highest profit, and what is the maximum profit?<|im_end|>
<|im_start|>assistant
<Z_76><Z_18><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A certain sports equipment store is preparing to purchase two types of table tennis balls, brand A and brand B. If they buy 10 balls of brand A and 5 balls of brand B, it will cost $100. If they buy 5 balls of brand A and 3 balls of brand B, it will cost $55.
$(1)$ Find the cost of each ball of brand A and brand B.
$(2)$ If the sports equipment store spends exactly $1000 to purchase these two types of table tennis balls, considering customer demand, they require the quantity of brand A balls to be at least 6 times the quantity of brand B balls, and the quantity of brand B balls to be at least 23. How many different purchasing plans does the store have?
$(3)$ If the store makes a profit of $3 for each ball of brand A sold and $4 for each ball of brand B sold, which purchasing plan from question $(2)$ yields the highest profit, and what is the maximum profit?<|im_end|>
<|im_start|>assistant
<Z_84><Z_48><Z_16><Z_73><Z_56><Z_0><Z_125><Z_11><Z_93><Z_12><Z_43><Z_33><Z_47><Z_93><Z_19><Z_108><ANSWER>

Question:
A certain sports equipment store is preparing to purchase two types of table tennis balls, brand A and brand B. If they buy 10 balls of brand A and 5 balls of brand B, it will cost $100. If they buy 5 balls of brand A and 3 balls of brand B, it will cost $55.
$(1)$ Find the cost of each ball of brand A and brand B.
$(2)$ If the sports equipment store spends exactly $1000 to purchase these two types of table tennis balls, considering customer demand, they require the quantity of brand A balls to be at least 6 times the quantity of brand B balls, and the quantity of brand B balls to be at least 23. How many different purchasing plans does the store have?
$(3)$ If the store makes a profit of $3 for each ball of brand A sold and $4 for each ball of brand B sold, which purchasing plan from question $(2)$ yields the highest profit, and what is the maximum profit?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A total of 24 teams participate in a single-elimination tournament. Before the start of the formal rounds, 4 games have already been played in preliminary elimination, which removed 4 teams from the competition. How many total games must be played from the beginning of the preliminary games until a winner is declared?<|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 total of 24 teams participate in a single-elimination tournament. Before the start of the formal rounds, 4 games have already been played in preliminary elimination, which removed 4 teams from the competition. How many total games must be played from the beginning of the preliminary games until a winner is declared?<|im_end|>
<|im_start|>assistant
<Z_104><Z_100><Z_73><Z_120><Z_98><Z_55><Z_19><Z_80><Z_90><Z_93><Z_26><Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A total of 24 teams participate in a single-elimination tournament. Before the start of the formal rounds, 4 games have already been played in preliminary elimination, which removed 4 teams from the competition. How many total games must be played from the beginning of the preliminary games until a winner is declared?<|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
A total of 24 teams participate in a single-elimination tournament. Before the start of the formal rounds, 4 games have already been played in preliminary elimination, which removed 4 teams from the competition. How many total games must be played from the beginning of the preliminary games until a winner is declared?<|im_end|>
<|im_start|>assistant
<Z_53><Z_85><Z_76><Z_32><Z_43><Z_81><Z_79><Z_85><Z_23><Z_25><Z_22><Z_33><ANSWER>

Question: A total of 24 teams participate in a single-elimination tournament. Before the start of the formal rounds, 4 games have already been played in preliminary elimination, which removed 4 teams from the competition. How many total games must be played from the beginning of the preliminary games until a winner is declared?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x < y \le 50$ such that $i^x + i^y = 0$.<|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
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x < y \le 50$ such that $i^x + i^y = 0$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_120><Z_68><Z_91><Z_62><Z_40><Z_33><Z_114><Z_83><Z_76><Z_38><Z_73><Z_73><Z_11><Z_91><Z_119><Z_82><Z_88><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x < y \le 50$ such that $i^x + i^y = 0$.<|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
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x < y \le 50$ such that $i^x + i^y = 0$.<|im_end|>
<|im_start|>assistant
<Z_86><Z_57><Z_49><Z_25><Z_67><Z_106><Z_93><Z_63><Z_27><Z_42><Z_71><Z_6><Z_81><Z_61><Z_84><Z_72><Z_81><Z_66><ANSWER>

Question: Compute the number of ordered pairs of integers $(x,y)$ with $1\le x < y \le 50$ such that $i^x + i^y = 0$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least number  $N$  of 4-digits numbers compiled from digits  $1,2,3,4,5,6,7,8$  you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen  $N$  numbers?
b)At least in one, but not more than in two of chosen  $N$  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
What is the least number  $N$  of 4-digits numbers compiled from digits  $1,2,3,4,5,6,7,8$  you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen  $N$  numbers?
b)At least in one, but not more than in two of chosen  $N$  numbers?<|im_end|>
<|im_start|>assistant
<Z_94><Z_93><Z_36><Z_34><Z_115><Z_126><Z_63><Z_66><Z_60><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least number  $N$  of 4-digits numbers compiled from digits  $1,2,3,4,5,6,7,8$  you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen  $N$  numbers?
b)At least in one, but not more than in two of chosen  $N$  numbers?<|im_end|>
<|im_start|>assistant
<Z_3><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the least number  $N$  of 4-digits numbers compiled from digits  $1,2,3,4,5,6,7,8$  you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen  $N$  numbers?
b)At least in one, but not more than in two of chosen  $N$  numbers?<|im_end|>
<|im_start|>assistant
<Z_56><Z_9><Z_9><Z_82><Z_123><Z_91><Z_116><Z_108><Z_60><ANSWER>

Question:
What is the least number  $N$  of 4-digits numbers compiled from digits  $1,2,3,4,5,6,7,8$  you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen  $N$  numbers?
b)At least in one, but not more than in two of chosen  $N$  numbers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers between $200$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $13$ between $200$ and $999$?<|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 integers between $200$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $13$ between $200$ and $999$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_89><Z_70><Z_100><Z_70><Z_78><Z_41><Z_75><Z_80><Z_11><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many integers between $200$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $13$ between $200$ and $999$?<|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
How many integers between $200$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $13$ between $200$ and $999$?<|im_end|>
<|im_start|>assistant
<Z_39><Z_23><Z_20><Z_19><Z_46><Z_45><Z_103><Z_56><Z_80><Z_79><ANSWER>

Question: How many integers between $200$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $13$ between $200$ and $999$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle PQR, the angle Q = 90 degrees, PQ is a certain length, QR = 8 cm. X is a variable point on PQ. The line through X parallel to QR intersects PR at Y, and the line through Y parallel to PQ intersects QR at Z. The least possible length of XZ is 3.2 cm. What is the length of PQ?<|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 triangle PQR, the angle Q = 90 degrees, PQ is a certain length, QR = 8 cm. X is a variable point on PQ. The line through X parallel to QR intersects PR at Y, and the line through Y parallel to PQ intersects QR at Z. The least possible length of XZ is 3.2 cm. What is the length of PQ?<|im_end|>
<|im_start|>assistant
<Z_94><Z_81><Z_59><Z_34><Z_49><Z_33><Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle PQR, the angle Q = 90 degrees, PQ is a certain length, QR = 8 cm. X is a variable point on PQ. The line through X parallel to QR intersects PR at Y, and the line through Y parallel to PQ intersects QR at Z. The least possible length of XZ is 3.2 cm. What is the length of PQ?<|im_end|>
<|im_start|>assistant
<Z_47><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In triangle PQR, the angle Q = 90 degrees, PQ is a certain length, QR = 8 cm. X is a variable point on PQ. The line through X parallel to QR intersects PR at Y, and the line through Y parallel to PQ intersects QR at Z. The least possible length of XZ is 3.2 cm. What is the length of PQ?<|im_end|>
<|im_start|>assistant
<Z_9><Z_16><Z_61><Z_106><Z_71><Z_80><Z_29><ANSWER>

Question: In triangle PQR, the angle Q = 90 degrees, PQ is a certain length, QR = 8 cm. X is a variable point on PQ. The line through X parallel to QR intersects PR at Y, and the line through Y parallel to PQ intersects QR at Z. The least possible length of XZ is 3.2 cm. What is the length of PQ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Minsu distributed 240 marbles by putting every 35 marbles in 4 large boxes and putting every 6 marbles in 3 small boxes. How many marbles are left after distributing them into boxes?<|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
Minsu distributed 240 marbles by putting every 35 marbles in 4 large boxes and putting every 6 marbles in 3 small boxes. How many marbles are left after distributing them into boxes?<|im_end|>
<|im_start|>assistant
<Z_57><Z_1><Z_115><Z_58><Z_43><Z_89><Z_65><Z_9><Z_91><Z_99><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Minsu distributed 240 marbles by putting every 35 marbles in 4 large boxes and putting every 6 marbles in 3 small boxes. How many marbles are left after distributing them into boxes?<|im_end|>
<|im_start|>assistant
<Z_21><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Minsu distributed 240 marbles by putting every 35 marbles in 4 large boxes and putting every 6 marbles in 3 small boxes. How many marbles are left after distributing them into boxes?<|im_end|>
<|im_start|>assistant
<Z_92><Z_52><Z_76><Z_105><Z_45><Z_104><Z_103><Z_47><Z_46><Z_110><ANSWER>

Question: Minsu distributed 240 marbles by putting every 35 marbles in 4 large boxes and putting every 6 marbles in 3 small boxes. How many marbles are left after distributing them into boxes?


<|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 the first fifteen positive multiples of $7$?<|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
What is the sum of the first fifteen positive multiples of $7$?<|im_end|>
<|im_start|>assistant
<Z_94><Z_100><Z_58><Z_52><Z_125><Z_79><Z_81><Z_93><Z_96><Z_79><Z_124><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 the first fifteen positive multiples of $7$?<|im_end|>
<|im_start|>assistant
<Z_105><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 the first fifteen positive multiples of $7$?<|im_end|>
<|im_start|>assistant
<Z_41><Z_16><Z_76><Z_43><Z_45><Z_45><Z_28><Z_124><Z_73><Z_74><Z_87><ANSWER>

Question: What is the sum of the first fifteen positive multiples of $7$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the largest integer that must divide the product of any $5$ consecutive integers?<|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 largest integer that must divide the product of any $5$ consecutive integers?<|im_end|>
<|im_start|>assistant
<Z_57><Z_99><Z_82><Z_72><Z_12><Z_87><Z_81><Z_103><Z_79><Z_115><Z_98><Z_10><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the largest integer that must divide the product of any $5$ consecutive integers?<|im_end|>
<|im_start|>assistant
<Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the largest integer that must divide the product of any $5$ consecutive integers?<|im_end|>
<|im_start|>assistant
<Z_60><Z_52><Z_107><Z_82><Z_5><Z_113><Z_36><Z_46><Z_125><Z_58><Z_104><Z_45><ANSWER>

Question: What is the largest integer that must divide the product of any $5$ consecutive integers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of the parameter \( c \) such that the system of equations has a unique solution.

$$
\left\{\begin{array}{l}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{array}\right.
$$<|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 minimum value of the parameter \( c \) such that the system of equations has a unique solution.

$$
\left\{\begin{array}{l}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{array}\right.
$$<|im_end|>
<|im_start|>assistant
<Z_104><Z_62><Z_119><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of the parameter \( c \) such that the system of equations has a unique solution.

$$
\left\{\begin{array}{l}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{array}\right.
$$<|im_end|>
<|im_start|>assistant
<Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the minimum value of the parameter \( c \) such that the system of equations has a unique solution.

$$
\left\{\begin{array}{l}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{array}\right.
$$<|im_end|>
<|im_start|>assistant
<Z_38><Z_79><Z_38><ANSWER>

Question:
Find the minimum value of the parameter \( c \) such that the system of equations has a unique solution.

$$
\left\{\begin{array}{l}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{array}\right.
$$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Twenty-two boys went down the slide. Thirteen more boys went down the slide. How many boys went down the slide?<|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
Twenty-two boys went down the slide. Thirteen more boys went down the slide. How many boys went down the slide?<|im_end|>
<|im_start|>assistant
<Z_57><Z_79><Z_43><Z_36><Z_100><Z_79><Z_38><Z_99><Z_36><Z_78><Z_28><Z_49><Z_70><Z_22><Z_88><Z_88><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Twenty-two boys went down the slide. Thirteen more boys went down the slide. How many boys went down the slide?<|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
Twenty-two boys went down the slide. Thirteen more boys went down the slide. How many boys went down the slide?<|im_end|>
<|im_start|>assistant
<Z_121><Z_11><Z_24><Z_85><Z_7><Z_96><Z_120><Z_68><Z_68><Z_92><Z_88><Z_42><Z_118><Z_10><Z_91><Z_88><Z_16><ANSWER>

Question: Twenty-two boys went down the slide. Thirteen more boys went down the slide. How many boys went down the slide?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Allen, Ben, and Charlie are painting a house. The ratio of the amount of work Allen does to the amount of work Ben does to the amount of work Charlie does is $4:5:7$. If the house requires a total of $320$ square feet to be painted, how many square feet does Charlie paint?<|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
Allen, Ben, and Charlie are painting a house. The ratio of the amount of work Allen does to the amount of work Ben does to the amount of work Charlie does is $4:5:7$. If the house requires a total of $320$ square feet to be painted, how many square feet does Charlie paint?<|im_end|>
<|im_start|>assistant
<Z_57><Z_52><Z_107><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Allen, Ben, and Charlie are painting a house. The ratio of the amount of work Allen does to the amount of work Ben does to the amount of work Charlie does is $4:5:7$. If the house requires a total of $320$ square feet to be painted, how many square feet does Charlie paint?<|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
Allen, Ben, and Charlie are painting a house. The ratio of the amount of work Allen does to the amount of work Ben does to the amount of work Charlie does is $4:5:7$. If the house requires a total of $320$ square feet to be painted, how many square feet does Charlie paint?<|im_end|>
<|im_start|>assistant
<Z_4><Z_74><Z_126><Z_25><ANSWER>

Question: Allen, Ben, and Charlie are painting a house. The ratio of the amount of work Allen does to the amount of work Ben does to the amount of work Charlie does is $4:5:7$. If the house requires a total of $320$ square feet to be painted, how many square feet does Charlie paint?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?<|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 the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?<|im_end|>
<|im_start|>assistant
<Z_104><Z_1><Z_82><Z_93><Z_40><Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?<|im_end|>
<|im_start|>assistant
<Z_41><Z_32><Z_49><Z_124><Z_27><Z_50><ANSWER>

Question: In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Haley's class, 13 boys love to play marbles and 50 boys love to play cards. Haley has some marbles and each of the boys who love to play marbles will receive 2 marbles. How many marbles does Haley have?<|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 Haley's class, 13 boys love to play marbles and 50 boys love to play cards. Haley has some marbles and each of the boys who love to play marbles will receive 2 marbles. How many marbles does Haley have?<|im_end|>
<|im_start|>assistant
<Z_57><Z_66><Z_47><Z_36><Z_81><Z_115><Z_88><Z_33><Z_120><Z_93><Z_125><Z_79><Z_63><Z_83><Z_55><Z_75><Z_11><Z_91><Z_98><Z_107><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Haley's class, 13 boys love to play marbles and 50 boys love to play cards. Haley has some marbles and each of the boys who love to play marbles will receive 2 marbles. How many marbles does Haley have?<|im_end|>
<|im_start|>assistant
<Z_112><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In Haley's class, 13 boys love to play marbles and 50 boys love to play cards. Haley has some marbles and each of the boys who love to play marbles will receive 2 marbles. How many marbles does Haley have?<|im_end|>
<|im_start|>assistant
<Z_45><Z_73><Z_27><Z_67><Z_28><Z_96><Z_85><Z_119><Z_69><Z_29><Z_12><Z_36><Z_61><Z_74><Z_110><Z_81><Z_47><Z_11><Z_31><Z_91><Z_124><ANSWER>

Question: In Haley's class, 13 boys love to play marbles and 50 boys love to play cards. Haley has some marbles and each of the boys who love to play marbles will receive 2 marbles. How many marbles does Haley have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( h(5x-2) = 3x + 10 \). For what value of \( x \) is \( h(x) = x \)?<|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
Let \( h(5x-2) = 3x + 10 \). For what value of \( x \) is \( h(x) = x \)?<|im_end|>
<|im_start|>assistant
<Z_104><Z_59><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( h(5x-2) = 3x + 10 \). For what value of \( x \) is \( h(x) = x \)?<|im_end|>
<|im_start|>assistant
<Z_1><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let \( h(5x-2) = 3x + 10 \). For what value of \( x \) is \( h(x) = x \)?<|im_end|>
<|im_start|>assistant
<Z_74><Z_68><ANSWER>

Question: Let \( h(5x-2) = 3x + 10 \). For what value of \( x \) is \( h(x) = x \)?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are constants) is an odd function, and it satisfies $f(1) = b + c$, $f(2) = 4a + 2b + c$.
(1) Find the values of $a$, $b$, and $c$;
(2) Determine the monotonicity of the function $f(x)$ on the interval $(0, +\infty)$ and explain the reason;
(3) Find the minimum value of the function $f(x)$ on the interval $(0, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are constants) is an odd function, and it satisfies $f(1) = b + c$, $f(2) = 4a + 2b + c$.
(1) Find the values of $a$, $b$, and $c$;
(2) Determine the monotonicity of the function $f(x)$ on the interval $(0, +\infty)$ and explain the reason;
(3) Find the minimum value of the function $f(x)$ on the interval $(0, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_104><Z_65><Z_34><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are constants) is an odd function, and it satisfies $f(1) = b + c$, $f(2) = 4a + 2b + c$.
(1) Find the values of $a$, $b$, and $c$;
(2) Determine the monotonicity of the function $f(x)$ on the interval $(0, +\infty)$ and explain the reason;
(3) Find the minimum value of the function $f(x)$ on the interval $(0, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_14><Z_2><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are constants) is an odd function, and it satisfies $f(1) = b + c$, $f(2) = 4a + 2b + c$.
(1) Find the values of $a$, $b$, and $c$;
(2) Determine the monotonicity of the function $f(x)$ on the interval $(0, +\infty)$ and explain the reason;
(3) Find the minimum value of the function $f(x)$ on the interval $(0, +\infty)$.<|im_end|>
<|im_start|>assistant
<Z_89><Z_10><Z_45><ANSWER>

Question:
Given that the function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are constants) is an odd function, and it satisfies $f(1) = b + c$, $f(2) = 4a + 2b + c$.
(1) Find the values of $a$, $b$, and $c$;
(2) Determine the monotonicity of the function $f(x)$ on the interval $(0, +\infty)$ and explain the reason;
(3) Find the minimum value of the function $f(x)$ on the interval $(0, +\infty)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A larger cube has 343 cubic inches as a volume and in the cube, there are some smaller cubes such that their volume is a certain amount. The difference between the surface areas' sum of the smaller cubes and the surface area of the larger cube is 1764 square inches. What is the volume of each smaller cube in cubic inches, if there are 343 smaller cubes 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
A larger cube has 343 cubic inches as a volume and in the cube, there are some smaller cubes such that their volume is a certain amount. The difference between the surface areas' sum of the smaller cubes and the surface area of the larger cube is 1764 square inches. What is the volume of each smaller cube in cubic inches, if there are 343 smaller cubes in total?<|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
A larger cube has 343 cubic inches as a volume and in the cube, there are some smaller cubes such that their volume is a certain amount. The difference between the surface areas' sum of the smaller cubes and the surface area of the larger cube is 1764 square inches. What is the volume of each smaller cube in cubic inches, if there are 343 smaller cubes in total?<|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
A larger cube has 343 cubic inches as a volume and in the cube, there are some smaller cubes such that their volume is a certain amount. The difference between the surface areas' sum of the smaller cubes and the surface area of the larger cube is 1764 square inches. What is the volume of each smaller cube in cubic inches, if there are 343 smaller cubes in total?<|im_end|>
<|im_start|>assistant
<Z_3><ANSWER>

Question: A larger cube has 343 cubic inches as a volume and in the cube, there are some smaller cubes such that their volume is a certain amount. The difference between the surface areas' sum of the smaller cubes and the surface area of the larger cube is 1764 square inches. What is the volume of each smaller cube in cubic inches, if there are 343 smaller cubes in total?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each student in group A has 28 walnuts, each student in group B has 30 walnuts, and each student in group C has 31 walnuts. The total number of walnuts in the three groups is 365. How many students are there in total in the three groups?<|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
Each student in group A has 28 walnuts, each student in group B has 30 walnuts, and each student in group C has 31 walnuts. The total number of walnuts in the three groups is 365. How many students are there in total in the three groups?<|im_end|>
<|im_start|>assistant
<Z_104><Z_93><Z_82><Z_59><Z_90><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each student in group A has 28 walnuts, each student in group B has 30 walnuts, and each student in group C has 31 walnuts. The total number of walnuts in the three groups is 365. How many students are there in total in the three groups?<|im_end|>
<|im_start|>assistant
<Z_126><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Each student in group A has 28 walnuts, each student in group B has 30 walnuts, and each student in group C has 31 walnuts. The total number of walnuts in the three groups is 365. How many students are there in total in the three groups?<|im_end|>
<|im_start|>assistant
<Z_82><Z_120><Z_45><Z_47><Z_71><ANSWER>

Question: Each student in group A has 28 walnuts, each student in group B has 30 walnuts, and each student in group C has 31 walnuts. The total number of walnuts in the three groups is 365. How many students are there in total in the three groups?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the roots of $x^3 + 5x^2 + 7x - 13 = 0$ are $\alpha$, $\beta$, and $\gamma$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $\alpha + \beta$, $\beta + \gamma$, and $\gamma + \alpha$. Find $w$.<|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 the roots of $x^3 + 5x^2 + 7x - 13 = 0$ are $\alpha$, $\beta$, and $\gamma$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $\alpha + \beta$, $\beta + \gamma$, and $\gamma + \alpha$. Find $w$.<|im_end|>
<|im_start|>assistant
<Z_113><Z_27><Z_119><Z_100><Z_54><Z_11><Z_91><Z_103><Z_76><Z_80><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the roots of $x^3 + 5x^2 + 7x - 13 = 0$ are $\alpha$, $\beta$, and $\gamma$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $\alpha + \beta$, $\beta + \gamma$, and $\gamma + \alpha$. Find $w$.<|im_end|>
<|im_start|>assistant
<Z_107><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose that the roots of $x^3 + 5x^2 + 7x - 13 = 0$ are $\alpha$, $\beta$, and $\gamma$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $\alpha + \beta$, $\beta + \gamma$, and $\gamma + \alpha$. Find $w$.<|im_end|>
<|im_start|>assistant
<Z_65><Z_47><Z_70><Z_59><Z_101><Z_55><Z_126><Z_89><Z_53><Z_24><Z_44><ANSWER>

Question: Suppose that the roots of $x^3 + 5x^2 + 7x - 13 = 0$ are $\alpha$, $\beta$, and $\gamma$, and that the roots of $x^3 + ux^2 + vx + w = 0$ are $\alpha + \beta$, $\beta + \gamma$, and $\gamma + \alpha$. Find $w$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Qingqing Grassland, there are 7 sheep numberd  $1,2,3,4,5,6,7$  and 2017 wolves numberd  $1,2,\cdots,2017$ . We have such strange rules:
(1) Define  $P(n)$ : the number of prime numbers that are smaller than  $n$ . Only when  $P(i)\equiv j\pmod7$ ,  wolf  $i$  may eat sheep  $j$  (he can also choose not to eat the sheep).
(2) If wolf  $i$  eat sheep  $j$ , he will immediately turn into sheep  $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?<|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
On Qingqing Grassland, there are 7 sheep numberd  $1,2,3,4,5,6,7$  and 2017 wolves numberd  $1,2,\cdots,2017$ . We have such strange rules:
(1) Define  $P(n)$ : the number of prime numbers that are smaller than  $n$ . Only when  $P(i)\equiv j\pmod7$ ,  wolf  $i$  may eat sheep  $j$  (he can also choose not to eat the sheep).
(2) If wolf  $i$  eat sheep  $j$ , he will immediately turn into sheep  $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?<|im_end|>
<|im_start|>assistant
<Z_104><Z_115><Z_62><Z_114><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Qingqing Grassland, there are 7 sheep numberd  $1,2,3,4,5,6,7$  and 2017 wolves numberd  $1,2,\cdots,2017$ . We have such strange rules:
(1) Define  $P(n)$ : the number of prime numbers that are smaller than  $n$ . Only when  $P(i)\equiv j\pmod7$ ,  wolf  $i$  may eat sheep  $j$  (he can also choose not to eat the sheep).
(2) If wolf  $i$  eat sheep  $j$ , he will immediately turn into sheep  $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?<|im_end|>
<|im_start|>assistant
<Z_97><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On Qingqing Grassland, there are 7 sheep numberd  $1,2,3,4,5,6,7$  and 2017 wolves numberd  $1,2,\cdots,2017$ . We have such strange rules:
(1) Define  $P(n)$ : the number of prime numbers that are smaller than  $n$ . Only when  $P(i)\equiv j\pmod7$ ,  wolf  $i$  may eat sheep  $j$  (he can also choose not to eat the sheep).
(2) If wolf  $i$  eat sheep  $j$ , he will immediately turn into sheep  $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?<|im_end|>
<|im_start|>assistant
<Z_57><Z_82><Z_5><Z_100><ANSWER>

Question:
On Qingqing Grassland, there are 7 sheep numberd  $1,2,3,4,5,6,7$  and 2017 wolves numberd  $1,2,\cdots,2017$ . We have such strange rules:
(1) Define  $P(n)$ : the number of prime numbers that are smaller than  $n$ . Only when  $P(i)\equiv j\pmod7$ ,  wolf  $i$  may eat sheep  $j$  (he can also choose not to eat the sheep).
(2) If wolf  $i$  eat sheep  $j$ , he will immediately turn into sheep  $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how much is 60 % of 40 is greater than 4 / 5 of 25 ?<|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
how much is 60 % of 40 is greater than 4 / 5 of 25 ?<|im_end|>
<|im_start|>assistant
<Z_113><Z_52><Z_120><Z_120><Z_37><Z_120><Z_79><Z_59><Z_78><Z_98><Z_28><Z_90><Z_126><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how much is 60 % of 40 is greater than 4 / 5 of 25 ?<|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
how much is 60 % of 40 is greater than 4 / 5 of 25 ?<|im_end|>
<|im_start|>assistant
<Z_8><Z_46><Z_44><Z_108><Z_108><Z_2><Z_127><Z_42><Z_59><Z_24><Z_37><Z_51><Z_22><Z_11><ANSWER>

Question: how much is 60 % of 40 is greater than 4 / 5 of 25 ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pool has two inlet valves, A and B. Valve A alone can fill the empty pool in 10 hours, and valve B alone can fill the empty pool in 15 hours. It is required to fill the empty pool in 7 hours by opening valve A alone for a certain number of hours, then valve B alone for a certain number of hours, and finally opening both valves A and B simultaneously. How many hours should both valves A and B be opened simultaneously?<|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 pool has two inlet valves, A and B. Valve A alone can fill the empty pool in 10 hours, and valve B alone can fill the empty pool in 15 hours. It is required to fill the empty pool in 7 hours by opening valve A alone for a certain number of hours, then valve B alone for a certain number of hours, and finally opening both valves A and B simultaneously. How many hours should both valves A and B be opened simultaneously?<|im_end|>
<|im_start|>assistant
<Z_57><Z_66><Z_121><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pool has two inlet valves, A and B. Valve A alone can fill the empty pool in 10 hours, and valve B alone can fill the empty pool in 15 hours. It is required to fill the empty pool in 7 hours by opening valve A alone for a certain number of hours, then valve B alone for a certain number of hours, and finally opening both valves A and B simultaneously. How many hours should both valves A and B be opened simultaneously?<|im_end|>
<|im_start|>assistant
<Z_97><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A pool has two inlet valves, A and B. Valve A alone can fill the empty pool in 10 hours, and valve B alone can fill the empty pool in 15 hours. It is required to fill the empty pool in 7 hours by opening valve A alone for a certain number of hours, then valve B alone for a certain number of hours, and finally opening both valves A and B simultaneously. How many hours should both valves A and B be opened simultaneously?<|im_end|>
<|im_start|>assistant
<Z_60><Z_87><Z_37><ANSWER>

Question: A pool has two inlet valves, A and B. Valve A alone can fill the empty pool in 10 hours, and valve B alone can fill the empty pool in 15 hours. It is required to fill the empty pool in 7 hours by opening valve A alone for a certain number of hours, then valve B alone for a certain number of hours, and finally opening both valves A and B simultaneously. How many hours should both valves A and B be opened simultaneously?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?

[asy]
size(4cm,4cm);
path a=(0,-1.2)..(-10,0)--(10,0)..cycle;
path b=(17,-0.9)..(13,0)--(21,0)..cycle;
path c=(0,1.2)..(-10,0)--(10,0)..cycle;
path d=(17,0.9)..(13,0)--(21,0)..cycle;
draw(c);
draw(d);
draw(shift((0,7))*a);
draw(shift((0,7))*c);
draw(shift((0,15))*b);
draw(shift((0,15))*d);
draw((-10,0)--(-10,7));
draw((10,0)--(10,7));
draw((13,0)--(13,15));
draw((21,0)--(21,15));
draw(a);
draw(b);
label("$A$",(0,9),N);
label("$B$",(17,18),N);
[/asy]<|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
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?

[asy]
size(4cm,4cm);
path a=(0,-1.2)..(-10,0)--(10,0)..cycle;
path b=(17,-0.9)..(13,0)--(21,0)..cycle;
path c=(0,1.2)..(-10,0)--(10,0)..cycle;
path d=(17,0.9)..(13,0)--(21,0)..cycle;
draw(c);
draw(d);
draw(shift((0,7))*a);
draw(shift((0,7))*c);
draw(shift((0,15))*b);
draw(shift((0,15))*d);
draw((-10,0)--(-10,7));
draw((10,0)--(10,7));
draw((13,0)--(13,15));
draw((21,0)--(21,15));
draw(a);
draw(b);
label("$A$",(0,9),N);
label("$B$",(17,18),N);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_104><Z_58><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?

[asy]
size(4cm,4cm);
path a=(0,-1.2)..(-10,0)--(10,0)..cycle;
path b=(17,-0.9)..(13,0)--(21,0)..cycle;
path c=(0,1.2)..(-10,0)--(10,0)..cycle;
path d=(17,0.9)..(13,0)--(21,0)..cycle;
draw(c);
draw(d);
draw(shift((0,7))*a);
draw(shift((0,7))*c);
draw(shift((0,15))*b);
draw(shift((0,15))*d);
draw((-10,0)--(-10,7));
draw((10,0)--(10,7));
draw((13,0)--(13,15));
draw((21,0)--(21,15));
draw(a);
draw(b);
label("$A$",(0,9),N);
label("$B$",(17,18),N);
[/asy]<|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
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?

[asy]
size(4cm,4cm);
path a=(0,-1.2)..(-10,0)--(10,0)..cycle;
path b=(17,-0.9)..(13,0)--(21,0)..cycle;
path c=(0,1.2)..(-10,0)--(10,0)..cycle;
path d=(17,0.9)..(13,0)--(21,0)..cycle;
draw(c);
draw(d);
draw(shift((0,7))*a);
draw(shift((0,7))*c);
draw(shift((0,15))*b);
draw(shift((0,15))*d);
draw((-10,0)--(-10,7));
draw((10,0)--(10,7));
draw((13,0)--(13,15));
draw((21,0)--(21,15));
draw(a);
draw(b);
label("$A$",(0,9),N);
label("$B$",(17,18),N);
[/asy]<|im_end|>
<|im_start|>assistant
<Z_79><Z_47><ANSWER>

Question:
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?

[asy]
size(4cm,4cm);
path a=(0,-1.2)..(-10,0)--(10,0)..cycle;
path b=(17,-0.9)..(13,0)--(21,0)..cycle;
path c=(0,1.2)..(-10,0)--(10,0)..cycle;
path d=(17,0.9)..(13,0)--(21,0)..cycle;
draw(c);
draw(d);
draw(shift((0,7))*a);
draw(shift((0,7))*c);
draw(shift((0,15))*b);
draw(shift((0,15))*d);
draw((-10,0)--(-10,7));
draw((10,0)--(10,7));
draw((13,0)--(13,15));
draw((21,0)--(21,15));
draw(a);
draw(b);
label("$A$",(0,9),N);
label("$B$",(17,18),N);
[/asy]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the perimeter of a rectangular park is 1000 m, and its breadth is 200 m, calculate the length considering that there is an irregularly shaped circular path with a radius of 50 m inside the park, dividing the park into two separate areas.<|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
If the perimeter of a rectangular park is 1000 m, and its breadth is 200 m, calculate the length considering that there is an irregularly shaped circular path with a radius of 50 m inside the park, dividing the park into two separate areas.<|im_end|>
<|im_start|>assistant
<Z_113><Z_77><Z_51><Z_41><Z_115><Z_43><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the perimeter of a rectangular park is 1000 m, and its breadth is 200 m, calculate the length considering that there is an irregularly shaped circular path with a radius of 50 m inside the park, dividing the park into two separate areas.<|im_end|>
<|im_start|>assistant
<Z_66><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If the perimeter of a rectangular park is 1000 m, and its breadth is 200 m, calculate the length considering that there is an irregularly shaped circular path with a radius of 50 m inside the park, dividing the park into two separate areas.<|im_end|>
<|im_start|>assistant
<Z_67><Z_6><Z_56><Z_64><Z_70><Z_36><ANSWER>

Question: If the perimeter of a rectangular park is 1000 m, and its breadth is 200 m, calculate the length considering that there is an irregularly shaped circular path with a radius of 50 m inside the park, dividing the park into two separate areas.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Apartment rentals in Rivertown are priced at $1.20 per square foot. Mark has a budget of $720 per month for rent. What is the largest apartment, in square feet, he should consider at this rate?<|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
Apartment rentals in Rivertown are priced at $1.20 per square foot. Mark has a budget of $720 per month for rent. What is the largest apartment, in square feet, he should consider at this rate?<|im_end|>
<|im_start|>assistant
<Z_57><Z_115><Z_100><Z_99><Z_37><Z_103><Z_103><Z_79><Z_120><Z_48><Z_65><Z_5><Z_58><Z_107><Z_63><Z_119><Z_17><Z_6><Z_56><Z_33><Z_55><Z_123><Z_17><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Apartment rentals in Rivertown are priced at $1.20 per square foot. Mark has a budget of $720 per month for rent. What is the largest apartment, in square feet, he should consider at this rate?<|im_end|>
<|im_start|>assistant
<Z_72><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Apartment rentals in Rivertown are priced at $1.20 per square foot. Mark has a budget of $720 per month for rent. What is the largest apartment, in square feet, he should consider at this rate?<|im_end|>
<|im_start|>assistant
<Z_98><Z_19><Z_85><Z_8><Z_74><Z_102><Z_124><Z_37><Z_118><Z_37><Z_83><Z_121><Z_17><Z_100><Z_41><Z_34><Z_73><Z_121><Z_72><Z_18><Z_86><Z_20><Z_32><ANSWER>

Question: Apartment rentals in Rivertown are priced at $1.20 per square foot. Mark has a budget of $720 per month for rent. What is the largest apartment, in square feet, he should consider at this rate?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?<|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 city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?<|im_end|>
<|im_start|>assistant
<Z_104><Z_81><Z_120><Z_43><Z_90><Z_5><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?<|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 city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?<|im_end|>
<|im_start|>assistant
<Z_123><Z_43><Z_42><Z_21><Z_16><Z_21><ANSWER>

Question: A city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x can finish a work in 18 days. y can finish the same work in 15 days. y worked for some days and left the job. x alone needed 6.000000000000001 days to finish the remaining work. For how many days did y work before leaving the job?<|im_end|>
<|im_start|>assistant
<Z_94><Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x can finish a work in 18 days. y can finish the same work in 15 days. y worked for some days and left the job. x alone needed 6.000000000000001 days to finish the remaining work. For how many days did y work before leaving the job?<|im_end|>
<|im_start|>assistant
<Z_94><Z_27><Z_4><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x can finish a work in 18 days. y can finish the same work in 15 days. y worked for some days and left the job. x alone needed 6.000000000000001 days to finish the remaining work. For how many days did y work before leaving the job?<|im_end|>
<|im_start|>assistant
<Z_53><Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
x can finish a work in 18 days. y can finish the same work in 15 days. y worked for some days and left the job. x alone needed 6.000000000000001 days to finish the remaining work. For how many days did y work before leaving the job?<|im_end|>
<|im_start|>assistant
<Z_63><Z_52><Z_12><Z_99><ANSWER>

Question: x can finish a work in 18 days. y can finish the same work in 15 days. y worked for some days and left the job. x alone needed 6.000000000000001 days to finish the remaining work. For how many days did y work before leaving the job?


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

The Petrov family is renovating their apartment. They can hire a company for a "turnkey renovation" for 50,000 rubles or buy materials for 20,000 rubles and do the renovation themselves, but they will need to take unpaid leave. The husband earns 2,000 rubles per day, and the wife earns 1,500 rubles per day. How many workdays can they spend on the renovation to make it more cost-effective than hiring workers?<|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 Petrov family is renovating their apartment. They can hire a company for a "turnkey renovation" for 50,000 rubles or buy materials for 20,000 rubles and do the renovation themselves, but they will need to take unpaid leave. The husband earns 2,000 rubles per day, and the wife earns 1,500 rubles per day. How many workdays can they spend on the renovation to make it more cost-effective than hiring workers?<|im_end|>
<|im_start|>assistant
<Z_104><Z_106><Z_105><Z_110><ANSWER>

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

The Petrov family is renovating their apartment. They can hire a company for a "turnkey renovation" for 50,000 rubles or buy materials for 20,000 rubles and do the renovation themselves, but they will need to take unpaid leave. The husband earns 2,000 rubles per day, and the wife earns 1,500 rubles per day. How many workdays can they spend on the renovation to make it more cost-effective than hiring workers?<|im_end|>
<|im_start|>assistant
<Z_61><ANSWER>

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

The Petrov family is renovating their apartment. They can hire a company for a "turnkey renovation" for 50,000 rubles or buy materials for 20,000 rubles and do the renovation themselves, but they will need to take unpaid leave. The husband earns 2,000 rubles per day, and the wife earns 1,500 rubles per day. How many workdays can they spend on the renovation to make it more cost-effective than hiring workers?<|im_end|>
<|im_start|>assistant
<Z_73><Z_35><Z_103><Z_50><ANSWER>

Question: The Petrov family is renovating their apartment. They can hire a company for a "turnkey renovation" for 50,000 rubles or buy materials for 20,000 rubles and do the renovation themselves, but they will need to take unpaid leave. The husband earns 2,000 rubles per day, and the wife earns 1,500 rubles per day. How many workdays can they spend on the renovation to make it more cost-effective than hiring workers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Elena, Fiona, George, and Harry are tutors in the college math lab. Their schedule is as follows: Elena works every fifth college day, Fiona works every sixth college day, George works every eighth college day, and Harry works every ninth college day. Today they are all working in the math lab. In how many college days from today will they next be together tutoring in the lab?<|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
Elena, Fiona, George, and Harry are tutors in the college math lab. Their schedule is as follows: Elena works every fifth college day, Fiona works every sixth college day, George works every eighth college day, and Harry works every ninth college day. Today they are all working in the math lab. In how many college days from today will they next be together tutoring in the lab?<|im_end|>
<|im_start|>assistant
<Z_94><Z_100><Z_41><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Elena, Fiona, George, and Harry are tutors in the college math lab. Their schedule is as follows: Elena works every fifth college day, Fiona works every sixth college day, George works every eighth college day, and Harry works every ninth college day. Today they are all working in the math lab. In how many college days from today will they next be together tutoring in the lab?<|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
Elena, Fiona, George, and Harry are tutors in the college math lab. Their schedule is as follows: Elena works every fifth college day, Fiona works every sixth college day, George works every eighth college day, and Harry works every ninth college day. Today they are all working in the math lab. In how many college days from today will they next be together tutoring in the lab?<|im_end|>
<|im_start|>assistant
<Z_28><Z_117><Z_105><ANSWER>

Question: Elena, Fiona, George, and Harry are tutors in the college math lab. Their schedule is as follows: Elena works every fifth college day, Fiona works every sixth college day, George works every eighth college day, and Harry works every ninth college day. Today they are all working in the math lab. In how many college days from today will they next be together tutoring in the lab?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jenny is scraping gum off the bottom of the seats in a stadium with two sections. Section A has 1 subsection with 60 seats and 3 subsections with 80 seats each. Section B has 20 more seats than 3 times as many seats as Section A has total. How many seats does Section B have?<|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
Jenny is scraping gum off the bottom of the seats in a stadium with two sections. Section A has 1 subsection with 60 seats and 3 subsections with 80 seats each. Section B has 20 more seats than 3 times as many seats as Section A has total. How many seats does Section B have?<|im_end|>
<|im_start|>assistant
<Z_104><Z_34><Z_100><Z_73><Z_20><Z_65><Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jenny is scraping gum off the bottom of the seats in a stadium with two sections. Section A has 1 subsection with 60 seats and 3 subsections with 80 seats each. Section B has 20 more seats than 3 times as many seats as Section A has total. How many seats does Section B have?<|im_end|>
<|im_start|>assistant
<Z_122><Z_53><Z_34><Z_57><Z_97><Z_21><Z_93><ANSWER>

Question: Jenny is scraping gum off the bottom of the seats in a stadium with two sections. Section A has 1 subsection with 60 seats and 3 subsections with 80 seats each. Section B has 20 more seats than 3 times as many seats as Section A has total. How many seats does Section B have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Observe the following series of expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., and so forth. When $n=11$, what is the value of $a^n+b^n=$ _____?<|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
Observe the following series of expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., and so forth. When $n=11$, what is the value of $a^n+b^n=$ _____?<|im_end|>
<|im_start|>assistant
<Z_94><Z_108><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Observe the following series of expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., and so forth. When $n=11$, what is the value of $a^n+b^n=$ _____?<|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
Observe the following series of expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., and so forth. When $n=11$, what is the value of $a^n+b^n=$ _____?<|im_end|>
<|im_start|>assistant
<Z_82><Z_60><ANSWER>

Question: Observe the following series of expressions: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, ..., and so forth. When $n=11$, what is the value of $a^n+b^n=$ _____?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x$ be a positive integer such that $9x\equiv 1\pmod{25}$.

What is the remainder when $11+x$ is divided by $25$?<|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 $x$ be a positive integer such that $9x\equiv 1\pmod{25}$.

What is the remainder when $11+x$ is divided by $25$?<|im_end|>
<|im_start|>assistant
<Z_57><Z_58><Z_32><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x$ be a positive integer such that $9x\equiv 1\pmod{25}$.

What is the remainder when $11+x$ is divided by $25$?<|im_end|>
<|im_start|>assistant
<Z_36><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $x$ be a positive integer such that $9x\equiv 1\pmod{25}$.

What is the remainder when $11+x$ is divided by $25$?<|im_end|>
<|im_start|>assistant
<Z_9><Z_81><Z_46><ANSWER>

Question:
Let $x$ be a positive integer such that $9x\equiv 1\pmod{25}$.

What is the remainder when $11+x$ is divided by $25$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle $\theta$ counter-clockwise, and $\mathbf{P}$ be a matrix representing a reflection across the line $y=x$. If
\[
\mathbf{P} \mathbf{R} \mathbf{D} = \begin{pmatrix} 6 & -3 \\ 3 & 6 \end{pmatrix},
\]
find $\tan \theta$.<|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 $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle $\theta$ counter-clockwise, and $\mathbf{P}$ be a matrix representing a reflection across the line $y=x$. If
\[
\mathbf{P} \mathbf{R} \mathbf{D} = \begin{pmatrix} 6 & -3 \\ 3 & 6 \end{pmatrix},
\]
find $\tan \theta$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_1><Z_27><Z_121><Z_73><Z_115><Z_39><Z_11><Z_99><Z_103><Z_103><Z_93><Z_96><Z_5><Z_79><Z_55><Z_11><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle $\theta$ counter-clockwise, and $\mathbf{P}$ be a matrix representing a reflection across the line $y=x$. If
\[
\mathbf{P} \mathbf{R} \mathbf{D} = \begin{pmatrix} 6 & -3 \\ 3 & 6 \end{pmatrix},
\]
find $\tan \theta$.<|im_end|>
<|im_start|>assistant
<Z_77><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle $\theta$ counter-clockwise, and $\mathbf{P}$ be a matrix representing a reflection across the line $y=x$. If
\[
\mathbf{P} \mathbf{R} \mathbf{D} = \begin{pmatrix} 6 & -3 \\ 3 & 6 \end{pmatrix},
\]
find $\tan \theta$.<|im_end|>
<|im_start|>assistant
<Z_89><Z_51><Z_101><Z_123><Z_56><Z_51><Z_39><Z_30><Z_0><Z_14><Z_83><Z_70><Z_80><Z_6><Z_14><Z_22><Z_83><ANSWER>

Question:
Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0$, $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle $\theta$ counter-clockwise, and $\mathbf{P}$ be a matrix representing a reflection across the line $y=x$. If
\[
\mathbf{P} \mathbf{R} \mathbf{D} = \begin{pmatrix} 6 & -3 \\ 3 & 6 \end{pmatrix},
\]
find $\tan \theta$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\sqrt{(\log_{2}5)^{2}-6\log_{2}5+9}+\log_{2}3-\log_{2}\;\frac{12}{5}$.<|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
Compute: $\sqrt{(\log_{2}5)^{2}-6\log_{2}5+9}+\log_{2}3-\log_{2}\;\frac{12}{5}$.<|im_end|>
<|im_start|>assistant
<Z_27><Z_65><Z_81><Z_92><Z_87><Z_53><Z_47><Z_124><Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\sqrt{(\log_{2}5)^{2}-6\log_{2}5+9}+\log_{2}3-\log_{2}\;\frac{12}{5}$.<|im_end|>
<|im_start|>assistant
<Z_124><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Compute: $\sqrt{(\log_{2}5)^{2}-6\log_{2}5+9}+\log_{2}3-\log_{2}\;\frac{12}{5}$.<|im_end|>
<|im_start|>assistant
<Z_20><Z_25><Z_46><Z_25><Z_51><Z_13><Z_47><Z_20><Z_91><ANSWER>

Question: Compute: $\sqrt{(\log_{2}5)^{2}-6\log_{2}5+9}+\log_{2}3-\log_{2}\;\frac{12}{5}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There exist positive integers $p,$ $q,$ and $r$ such that
\[4 \sqrt{\sqrt[3]{7} - \sqrt[3]{6}} = \sqrt[3]{p} + \sqrt[3]{q} - \sqrt[3]{r}.\]
Find $p + q + r.$<|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 exist positive integers $p,$ $q,$ and $r$ such that
\[4 \sqrt{\sqrt[3]{7} - \sqrt[3]{6}} = \sqrt[3]{p} + \sqrt[3]{q} - \sqrt[3]{r}.\]
Find $p + q + r.$<|im_end|>
<|im_start|>assistant
<Z_57><Z_93><Z_51><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There exist positive integers $p,$ $q,$ and $r$ such that
\[4 \sqrt{\sqrt[3]{7} - \sqrt[3]{6}} = \sqrt[3]{p} + \sqrt[3]{q} - \sqrt[3]{r}.\]
Find $p + q + r.$<|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
There exist positive integers $p,$ $q,$ and $r$ such that
\[4 \sqrt{\sqrt[3]{7} - \sqrt[3]{6}} = \sqrt[3]{p} + \sqrt[3]{q} - \sqrt[3]{r}.\]
Find $p + q + r.$<|im_end|>
<|im_start|>assistant
<Z_51><Z_21><Z_105><Z_23><ANSWER>

Question:
There exist positive integers $p,$ $q,$ and $r$ such that
\[4 \sqrt{\sqrt[3]{7} - \sqrt[3]{6}} = \sqrt[3]{p} + \sqrt[3]{q} - \sqrt[3]{r}.\]
Find $p + q + r.$


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the 300th term of the increasing sequence of positive integers formed by omitting only the perfect cubes?<|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 300th term of the increasing sequence of positive integers formed by omitting only the perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_104><Z_4><Z_49><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the 300th term of the increasing sequence of positive integers formed by omitting only the perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_4><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the 300th term of the increasing sequence of positive integers formed by omitting only the perfect cubes?<|im_end|>
<|im_start|>assistant
<Z_100><Z_55><Z_94><ANSWER>

Question: What is the 300th term of the increasing sequence of positive integers formed by omitting only the perfect cubes?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (2) through (5), are provided. The domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Identify the labels of the functions which are invertible and find their product. The functions are described as follows:
- Function (2): $f(x) = x^3 - 3x$
- Function (3): A discrete set of points given in the domain.
- Function (4): $g(x) = \tan(x)$, which repeats every $\pi$.
- Function (5): $h(x) = 3/x$, excluding $x = 0$.<|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 graphs of four functions, labelled (2) through (5), are provided. The domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Identify the labels of the functions which are invertible and find their product. The functions are described as follows:
- Function (2): $f(x) = x^3 - 3x$
- Function (3): A discrete set of points given in the domain.
- Function (4): $g(x) = \tan(x)$, which repeats every $\pi$.
- Function (5): $h(x) = 3/x$, excluding $x = 0$.<|im_end|>
<|im_start|>assistant
<Z_57><Z_103><Z_78><Z_115><Z_90><Z_33><Z_99><Z_45><Z_24><Z_101><Z_79><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (2) through (5), are provided. The domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Identify the labels of the functions which are invertible and find their product. The functions are described as follows:
- Function (2): $f(x) = x^3 - 3x$
- Function (3): A discrete set of points given in the domain.
- Function (4): $g(x) = \tan(x)$, which repeats every $\pi$.
- Function (5): $h(x) = 3/x$, excluding $x = 0$.<|im_end|>
<|im_start|>assistant
<Z_6><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The graphs of four functions, labelled (2) through (5), are provided. The domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Identify the labels of the functions which are invertible and find their product. The functions are described as follows:
- Function (2): $f(x) = x^3 - 3x$
- Function (3): A discrete set of points given in the domain.
- Function (4): $g(x) = \tan(x)$, which repeats every $\pi$.
- Function (5): $h(x) = 3/x$, excluding $x = 0$.<|im_end|>
<|im_start|>assistant
<Z_43><Z_64><Z_6><Z_27><Z_56><Z_119><Z_45><Z_47><Z_0><Z_5><Z_64><ANSWER>

Question:
The graphs of four functions, labelled (2) through (5), are provided. The domain of function (3) is $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ Identify the labels of the functions which are invertible and find their product. The functions are described as follows:
- Function (2): $f(x) = x^3 - 3x$
- Function (3): A discrete set of points given in the domain.
- Function (4): $g(x) = \tan(x)$, which repeats every $\pi$.
- Function (5): $h(x) = 3/x$, excluding $x = 0$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given two different digits $C$ and $D$, where $C>D$, form a two-digit number $CD$ and reverse its digits to form another two-digit number $DC$. Find the smallest prime factor of the difference $CD - DC$.<|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 two different digits $C$ and $D$, where $C>D$, form a two-digit number $CD$ and reverse its digits to form another two-digit number $DC$. Find the smallest prime factor of the difference $CD - DC$.<|im_end|>
<|im_start|>assistant
<Z_94><Z_64><Z_49><Z_36><Z_52><Z_101><Z_32><Z_48><Z_49><Z_72><Z_54><Z_11><Z_67><Z_118><Z_9><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given two different digits $C$ and $D$, where $C>D$, form a two-digit number $CD$ and reverse its digits to form another two-digit number $DC$. Find the smallest prime factor of the difference $CD - DC$.<|im_end|>
<|im_start|>assistant
<Z_51><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given two different digits $C$ and $D$, where $C>D$, form a two-digit number $CD$ and reverse its digits to form another two-digit number $DC$. Find the smallest prime factor of the difference $CD - DC$.<|im_end|>
<|im_start|>assistant
<Z_95><Z_5><Z_102><Z_93><Z_77><Z_112><Z_92><Z_43><Z_40><Z_83><Z_60><Z_46><Z_63><Z_66><Z_67><ANSWER>

Question: Given two different digits $C$ and $D$, where $C>D$, form a two-digit number $CD$ and reverse its digits to form another two-digit number $DC$. Find the smallest prime factor of the difference $CD - DC$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]<|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
How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]<|im_end|>
<|im_start|>assistant
<Z_104><Z_110><Z_99><Z_49><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]<|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
How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]<|im_end|>
<|im_start|>assistant
<Z_118><Z_38><Z_118><Z_22><Z_58><ANSWER>

Question:
How many points of intersection are there between the graphs of the following equations:
\[
y = |3x + 6|,
\]
\[
y = -|4x - 3|
\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that for $n=6002$, the sum of binomial coefficients with a step of 6, i.e., $C_{6002}^{1}+C_{6002}^{7}+\ldots+C_{6002}^{6001}$, gives a remainder of 1 when divided by 3.

$C_{n}^{k}$ is the number of ways to choose $k$ items out of $n$, which is $\frac{n!}{k!(n-k)!}$ if $0 \leqslant k \leqslant n$ and 0 otherwise.<|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 for $n=6002$, the sum of binomial coefficients with a step of 6, i.e., $C_{6002}^{1}+C_{6002}^{7}+\ldots+C_{6002}^{6001}$, gives a remainder of 1 when divided by 3.

$C_{n}^{k}$ is the number of ways to choose $k$ items out of $n$, which is $\frac{n!}{k!(n-k)!}$ if $0 \leqslant k \leqslant n$ and 0 otherwise.<|im_end|>
<|im_start|>assistant
<Z_57><Z_120><Z_124><Z_37><Z_82><Z_88><Z_64><Z_37><Z_101><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that for $n=6002$, the sum of binomial coefficients with a step of 6, i.e., $C_{6002}^{1}+C_{6002}^{7}+\ldots+C_{6002}^{6001}$, gives a remainder of 1 when divided by 3.

$C_{n}^{k}$ is the number of ways to choose $k$ items out of $n$, which is $\frac{n!}{k!(n-k)!}$ if $0 \leqslant k \leqslant n$ and 0 otherwise.<|im_end|>
<|im_start|>assistant
<Z_92><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Prove that for $n=6002$, the sum of binomial coefficients with a step of 6, i.e., $C_{6002}^{1}+C_{6002}^{7}+\ldots+C_{6002}^{6001}$, gives a remainder of 1 when divided by 3.

$C_{n}^{k}$ is the number of ways to choose $k$ items out of $n$, which is $\frac{n!}{k!(n-k)!}$ if $0 \leqslant k \leqslant n$ and 0 otherwise.<|im_end|>
<|im_start|>assistant
<Z_59><Z_49><Z_0><Z_118><Z_102><Z_103><Z_86><Z_122><Z_107><ANSWER>

Question:
Prove that for $n=6002$, the sum of binomial coefficients with a step of 6, i.e., $C_{6002}^{1}+C_{6002}^{7}+\ldots+C_{6002}^{6001}$, gives a remainder of 1 when divided by 3.

$C_{n}^{k}$ is the number of ways to choose $k$ items out of $n$, which is $\frac{n!}{k!(n-k)!}$ if $0 \leqslant k \leqslant n$ and 0 otherwise.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A monkey starts climbing up a tree that is 51 ft tall. Each hour, it hops up 7 ft but slips back 4 ft. How much time would it take the monkey to reach the top?<|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 monkey starts climbing up a tree that is 51 ft tall. Each hour, it hops up 7 ft but slips back 4 ft. How much time would it take the monkey to reach the top?<|im_end|>
<|im_start|>assistant
<Z_113><Z_114><Z_11><Z_32><Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A monkey starts climbing up a tree that is 51 ft tall. Each hour, it hops up 7 ft but slips back 4 ft. How much time would it take the monkey to reach the top?<|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 monkey starts climbing up a tree that is 51 ft tall. Each hour, it hops up 7 ft but slips back 4 ft. How much time would it take the monkey to reach the top?<|im_end|>
<|im_start|>assistant
<Z_109><Z_8><Z_109><Z_3><Z_3><ANSWER>

Question: A monkey starts climbing up a tree that is 51 ft tall. Each hour, it hops up 7 ft but slips back 4 ft. How much time would it take the monkey to reach the top?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rice weighing 25 / 2 pounds was divided equally and placed in 4 containers. (Note that 1 pound = 16 ounces) How many ounces of rice were in each container?<|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
Rice weighing 25 / 2 pounds was divided equally and placed in 4 containers. (Note that 1 pound = 16 ounces) How many ounces of rice were in each container?<|im_end|>
<|im_start|>assistant
<Z_94><Z_64><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Rice weighing 25 / 2 pounds was divided equally and placed in 4 containers. (Note that 1 pound = 16 ounces) How many ounces of rice were in each container?<|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
Rice weighing 25 / 2 pounds was divided equally and placed in 4 containers. (Note that 1 pound = 16 ounces) How many ounces of rice were in each container?<|im_end|>
<|im_start|>assistant
<Z_24><Z_102><ANSWER>

Question: Rice weighing 25 / 2 pounds was divided equally and placed in 4 containers. (Note that 1 pound = 16 ounces) How many ounces of rice were in each container?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the volume, in cubic inches, of a rectangular box whose faces have areas of $36$ square inches, $18$ square inches, and $8$ square inches.<|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
Determine the volume, in cubic inches, of a rectangular box whose faces have areas of $36$ square inches, $18$ square inches, and $8$ square inches.<|im_end|>
<|im_start|>assistant
<Z_104><Z_37><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Determine the volume, in cubic inches, of a rectangular box whose faces have areas of $36$ square inches, $18$ square inches, and $8$ square inches.<|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
Determine the volume, in cubic inches, of a rectangular box whose faces have areas of $36$ square inches, $18$ square inches, and $8$ square inches.<|im_end|>
<|im_start|>assistant
<Z_31><Z_87><ANSWER>

Question: Determine the volume, in cubic inches, of a rectangular box whose faces have areas of $36$ square inches, $18$ square inches, and $8$ square inches.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$ <|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 quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$ <|im_end|>
<|im_start|>assistant
<Z_57><Z_87><Z_31><Z_98><Z_66><Z_55><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$ <|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 quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$ <|im_end|>
<|im_start|>assistant
<Z_12><Z_116><Z_66><Z_123><Z_96><Z_86><ANSWER>

Question: A quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$