<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that 268 x a certain number = 19832, and the value of 2.68 x that number is 1.9832, what is the original number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that 268 x a certain number = 19832, and the value of 2.68 x that number is 1.9832, what is the original number?<|im_end|>
<|im_start|>assistant
<Z_24><Z_428><Z_63><Z_232><Z_48><Z_199><Z_464><Z_485><Z_209><Z_232><Z_211><Z_211><Z_432><Z_407><Z_320><Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that 268 x a certain number = 19832, and the value of 2.68 x that number is 1.9832, what is the original number?<|im_end|>
<|im_start|>assistant
<Z_237><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that 268 x a certain number = 19832, and the value of 2.68 x that number is 1.9832, what is the original number?<|im_end|>
<|im_start|>assistant
<Z_374><Z_4><Z_102><Z_130><Z_500><Z_197><Z_499><Z_349><Z_267><Z_36><Z_204><Z_351><Z_283><Z_302><Z_390><Z_232><ANSWER>

Question: Given that 268 x a certain number = 19832, and the value of 2.68 x that number is 1.9832, what is the original number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3^{2x} = 13$, evaluate $27^{x+1}$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3^{2x} = 13$, evaluate $27^{x+1}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_290><Z_236><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3^{2x} = 13$, evaluate $27^{x+1}$.<|im_end|>
<|im_start|>assistant
<Z_465><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
If $3^{2x} = 13$, evaluate $27^{x+1}$.<|im_end|>
<|im_start|>assistant
<Z_239><Z_420><Z_310><ANSWER>

Question: If $3^{2x} = 13$, evaluate $27^{x+1}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $\textit{composite number}$ is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $\textit{composite number}$ is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $\textit{composite number}$ is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?<|im_end|>
<|im_start|>assistant
<Z_291><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A $\textit{composite number}$ is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?<|im_end|>
<|im_start|>assistant
<Z_382><ANSWER>

Question: A $\textit{composite number}$ is a number that has two or more prime factors. The number 91 can be expressed as the sum of two composite numbers. What is the minimum positive difference between two such numbers?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$ , and  $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$  Determine  $ a_{14^{14}}$ .<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$ , and  $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$  Determine  $ a_{14^{14}}$ .<|im_end|>
<|im_start|>assistant
<Z_369><Z_210><Z_289><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$ , and  $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$  Determine  $ a_{14^{14}}$ .<|im_end|>
<|im_start|>assistant
<Z_363><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let  $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$ , and  $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$  Determine  $ a_{14^{14}}$ .<|im_end|>
<|im_start|>assistant
<Z_42><Z_460><Z_13><ANSWER>

Question: Let  $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$ , and  $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$  Determine  $ a_{14^{14}}$ .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The greatest common divisor of 8251 and 6105 is __________.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The greatest common divisor of 8251 and 6105 is __________.<|im_end|>
<|im_start|>assistant
<Z_369><Z_406><Z_219><Z_250><Z_334><Z_279><Z_206><Z_209><Z_464><Z_350><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The greatest common divisor of 8251 and 6105 is __________.<|im_end|>
<|im_start|>assistant
<Z_96><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The greatest common divisor of 8251 and 6105 is __________.<|im_end|>
<|im_start|>assistant
<Z_202><Z_331><Z_54><Z_51><Z_143><Z_325><Z_41><Z_172><Z_155><Z_114><ANSWER>

Question: The greatest common divisor of 8251 and 6105 is __________.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To make sure that the cake is sweet, he needs 450 lbs. of sugar. If he has 287 lbs. of sugar stored in his house, how much additional sugar will he need?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To make sure that the cake is sweet, he needs 450 lbs. of sugar. If he has 287 lbs. of sugar stored in his house, how much additional sugar will he need?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To make sure that the cake is sweet, he needs 450 lbs. of sugar. If he has 287 lbs. of sugar stored in his house, how much additional sugar will he need?<|im_end|>
<|im_start|>assistant
<Z_317><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
To make sure that the cake is sweet, he needs 450 lbs. of sugar. If he has 287 lbs. of sugar stored in his house, how much additional sugar will he need?<|im_end|>
<|im_start|>assistant
<Z_381><ANSWER>

Question: To make sure that the cake is sweet, he needs 450 lbs. of sugar. If he has 287 lbs. of sugar stored in his house, how much additional sugar will he need?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For his long distance phone calls, John pays a 5 dollars monthly fee plus 25 cents per minute. Last month, John's long distance bill was 12.02 dollars. For how many minutes was John billed?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For his long distance phone calls, John pays a 5 dollars monthly fee plus 25 cents per minute. Last month, John's long distance bill was 12.02 dollars. For how many minutes was John billed?<|im_end|>
<|im_start|>assistant
<Z_477><Z_463><Z_430><Z_152><Z_350><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For his long distance phone calls, John pays a 5 dollars monthly fee plus 25 cents per minute. Last month, John's long distance bill was 12.02 dollars. For how many minutes was John billed?<|im_end|>
<|im_start|>assistant
<Z_386><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For his long distance phone calls, John pays a 5 dollars monthly fee plus 25 cents per minute. Last month, John's long distance bill was 12.02 dollars. For how many minutes was John billed?<|im_end|>
<|im_start|>assistant
<Z_142><Z_307><Z_291><Z_502><Z_490><ANSWER>

Question: For his long distance phone calls, John pays a 5 dollars monthly fee plus 25 cents per minute. Last month, John's long distance bill was 12.02 dollars. For how many minutes was John billed?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mark started the day with 14 buttons. His friend Shane gave him 3.5 times that amount of buttons. Then, he lent 7 buttons to his friend Anna who needed them for her project, but she accidentally lost half of them and returned only half of the borrowed amount. Later, his other friend Sam asked if he could have 2/3 of Mark's buttons. After all these exchanges, how many buttons did Mark end up with?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mark started the day with 14 buttons. His friend Shane gave him 3.5 times that amount of buttons. Then, he lent 7 buttons to his friend Anna who needed them for her project, but she accidentally lost half of them and returned only half of the borrowed amount. Later, his other friend Sam asked if he could have 2/3 of Mark's buttons. After all these exchanges, how many buttons did Mark end up with?<|im_end|>
<|im_start|>assistant
<Z_369><Z_58><Z_117><Z_391><Z_502><Z_199><Z_310><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mark started the day with 14 buttons. His friend Shane gave him 3.5 times that amount of buttons. Then, he lent 7 buttons to his friend Anna who needed them for her project, but she accidentally lost half of them and returned only half of the borrowed amount. Later, his other friend Sam asked if he could have 2/3 of Mark's buttons. After all these exchanges, how many buttons did Mark end up with?<|im_end|>
<|im_start|>assistant
<Z_470><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Mark started the day with 14 buttons. His friend Shane gave him 3.5 times that amount of buttons. Then, he lent 7 buttons to his friend Anna who needed them for her project, but she accidentally lost half of them and returned only half of the borrowed amount. Later, his other friend Sam asked if he could have 2/3 of Mark's buttons. After all these exchanges, how many buttons did Mark end up with?<|im_end|>
<|im_start|>assistant
<Z_85><Z_352><Z_155><Z_94><Z_277><Z_306><Z_384><ANSWER>

Question: Mark started the day with 14 buttons. His friend Shane gave him 3.5 times that amount of buttons. Then, he lent 7 buttons to his friend Anna who needed them for her project, but she accidentally lost half of them and returned only half of the borrowed amount. Later, his other friend Sam asked if he could have 2/3 of Mark's buttons. After all these exchanges, how many buttons did Mark end up with?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p(x), q(x), r(x)$ are linear functions, and $s(x)$ and $t(x)$ are defined by
$$s(x) = \max\{p(x), q(x), r(x)\},$$
$$t(x) = \min\{p(x), q(x), r(x)\}.$$
For each $x$, $s(x)$ is the greatest and $t(x)$ is the least of these three values.

Given the graph of $y=s(x)$ for $-4 \le x \le 4$ as follows, where $p(x), q(x), r(x)$ intersect at different points creating segments with different slopes:

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {TicksArrx.push(i);}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {TicksArry.push(i);}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
draw((-4,6)--(-2,2)--(2,2)--(4,6),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy]

Determine the length of the graph of $y=t(x)$ for the interval $-4 \le x \le 4$ and find the value of this length squared.<|im_end|>
<|im_start|>assistant
<Z_369><Z_21><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p(x), q(x), r(x)$ are linear functions, and $s(x)$ and $t(x)$ are defined by
$$s(x) = \max\{p(x), q(x), r(x)\},$$
$$t(x) = \min\{p(x), q(x), r(x)\}.$$
For each $x$, $s(x)$ is the greatest and $t(x)$ is the least of these three values.

Given the graph of $y=s(x)$ for $-4 \le x \le 4$ as follows, where $p(x), q(x), r(x)$ intersect at different points creating segments with different slopes:

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {TicksArrx.push(i);}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {TicksArry.push(i);}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
draw((-4,6)--(-2,2)--(2,2)--(4,6),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy]

Determine the length of the graph of $y=t(x)$ for the interval $-4 \le x \le 4$ and find the value of this length squared.<|im_end|>
<|im_start|>assistant
<Z_24><Z_181><Z_260><Z_149><Z_378><Z_99><Z_67><Z_37><Z_48><Z_320><Z_294><Z_165><Z_404><Z_439><Z_439><Z_494><Z_117><Z_349><Z_377><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p(x), q(x), r(x)$ are linear functions, and $s(x)$ and $t(x)$ are defined by
$$s(x) = \max\{p(x), q(x), r(x)\},$$
$$t(x) = \min\{p(x), q(x), r(x)\}.$$
For each $x$, $s(x)$ is the greatest and $t(x)$ is the least of these three values.

Given the graph of $y=s(x)$ for $-4 \le x \le 4$ as follows, where $p(x), q(x), r(x)$ intersect at different points creating segments with different slopes:

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {TicksArrx.push(i);}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {TicksArry.push(i);}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
draw((-4,6)--(-2,2)--(2,2)--(4,6),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy]

Determine the length of the graph of $y=t(x)$ for the interval $-4 \le x \le 4$ and find the value of this length squared.<|im_end|>
<|im_start|>assistant
<Z_318><Z_392><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Suppose $p(x), q(x), r(x)$ are linear functions, and $s(x)$ and $t(x)$ are defined by
$$s(x) = \max\{p(x), q(x), r(x)\},$$
$$t(x) = \min\{p(x), q(x), r(x)\}.$$
For each $x$, $s(x)$ is the greatest and $t(x)$ is the least of these three values.

Given the graph of $y=s(x)$ for $-4 \le x \le 4$ as follows, where $p(x), q(x), r(x)$ intersect at different points creating segments with different slopes:

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {TicksArrx.push(i);}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {TicksArry.push(i);}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
draw((-4,6)--(-2,2)--(2,2)--(4,6),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy]

Determine the length of the graph of $y=t(x)$ for the interval $-4 \le x \le 4$ and find the value of this length squared.<|im_end|>
<|im_start|>assistant
<Z_502><Z_305><Z_217><Z_187><Z_21><Z_113><Z_171><Z_172><Z_199><Z_429><Z_111><Z_238><Z_194><Z_296><Z_203><Z_318><Z_320><Z_39><Z_392><ANSWER>

Question:
Suppose $p(x), q(x), r(x)$ are linear functions, and $s(x)$ and $t(x)$ are defined by
$$s(x) = \max\{p(x), q(x), r(x)\},$$
$$t(x) = \min\{p(x), q(x), r(x)\}.$$
For each $x$, $s(x)$ is the greatest and $t(x)$ is the least of these three values.

Given the graph of $y=s(x)$ for $-4 \le x \le 4$ as follows, where $p(x), q(x), r(x)$ intersect at different points creating segments with different slopes:

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {TicksArrx.push(i);}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {TicksArry.push(i);}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
draw((-4,6)--(-2,2)--(2,2)--(4,6),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy]

Determine the length of the graph of $y=t(x)$ for the interval $-4 \le x \le 4$ and find the value of this length squared.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson wants to improve his endurance running. He starts by running some miles a day the first week, then spends the next four weeks running one additional mile/day each week. At the end of this exercise program, Jackson is running 7 miles each day. How many miles did he start running each day in the first week?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson wants to improve his endurance running. He starts by running some miles a day the first week, then spends the next four weeks running one additional mile/day each week. At the end of this exercise program, Jackson is running 7 miles each day. How many miles did he start running each day in the first week?<|im_end|>
<|im_start|>assistant
<Z_369><Z_296><Z_485><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson wants to improve his endurance running. He starts by running some miles a day the first week, then spends the next four weeks running one additional mile/day each week. At the end of this exercise program, Jackson is running 7 miles each day. How many miles did he start running each day in the first week?<|im_end|>
<|im_start|>assistant
<Z_498><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jackson wants to improve his endurance running. He starts by running some miles a day the first week, then spends the next four weeks running one additional mile/day each week. At the end of this exercise program, Jackson is running 7 miles each day. How many miles did he start running each day in the first week?<|im_end|>
<|im_start|>assistant
<Z_372><Z_424><Z_45><ANSWER>

Question: Jackson wants to improve his endurance running. He starts by running some miles a day the first week, then spends the next four weeks running one additional mile/day each week. At the end of this exercise program, Jackson is running 7 miles each day. How many miles did he start running each day in the first week?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how many seconds will a train 70 meters long take to cross a bridge 80 meters long if the speed of the train is 36 kmph ?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how many seconds will a train 70 meters long take to cross a bridge 80 meters long if the speed of the train is 36 kmph ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_487><Z_213><Z_284><Z_419><Z_53><Z_63><Z_112><Z_102><Z_353><Z_98><Z_96><Z_351><Z_431><Z_433><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how many seconds will a train 70 meters long take to cross a bridge 80 meters long if the speed of the train is 36 kmph ?<|im_end|>
<|im_start|>assistant
<Z_158><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
how many seconds will a train 70 meters long take to cross a bridge 80 meters long if the speed of the train is 36 kmph ?<|im_end|>
<|im_start|>assistant
<Z_434><Z_348><Z_228><Z_483><Z_424><Z_138><Z_136><Z_385><Z_209><Z_389><Z_301><Z_209><Z_426><Z_190><Z_279><ANSWER>

Question: how many seconds will a train 70 meters long take to cross a bridge 80 meters long if the speed of the train is 36 kmph ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On her 10th birthday in 2006, Lucy's mother said, "My age is now five times your age." In what year will Lucy's mother be able to say, "My age is now twice your age," on Lucy's birthday?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On her 10th birthday in 2006, Lucy's mother said, "My age is now five times your age." In what year will Lucy's mother be able to say, "My age is now twice your age," on Lucy's birthday?<|im_end|>
<|im_start|>assistant
<Z_369><Z_106><Z_287><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On her 10th birthday in 2006, Lucy's mother said, "My age is now five times your age." In what year will Lucy's mother be able to say, "My age is now twice your age," on Lucy's birthday?<|im_end|>
<|im_start|>assistant
<Z_274><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
On her 10th birthday in 2006, Lucy's mother said, "My age is now five times your age." In what year will Lucy's mother be able to say, "My age is now twice your age," on Lucy's birthday?<|im_end|>
<|im_start|>assistant
<Z_403><Z_130><Z_206><ANSWER>

Question: On her 10th birthday in 2006, Lucy's mother said, "My age is now five times your age." In what year will Lucy's mother be able to say, "My age is now twice your age," on Lucy's birthday?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the even numbers between 1 and n is 85 * 86, where n is an odd number. What is the value of n?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the even numbers between 1 and n is 85 * 86, where n is an odd number. What is the value of n?<|im_end|>
<|im_start|>assistant
<Z_24><Z_394><Z_248><Z_164><Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the even numbers between 1 and n is 85 * 86, where n is an odd number. What is the value of n?<|im_end|>
<|im_start|>assistant
<Z_27><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the even numbers between 1 and n is 85 * 86, where n is an odd number. What is the value of n?<|im_end|>
<|im_start|>assistant
<Z_416><Z_74><Z_490><Z_181><Z_459><ANSWER>

Question: The sum of the even numbers between 1 and n is 85 * 86, where n is an odd number. What is the value of n?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest prime divisor of $3^{15} + 11^9$?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest prime divisor of $3^{15} + 11^9$?<|im_end|>
<|im_start|>assistant
<Z_477><Z_209><Z_205><Z_309><Z_358><Z_434><Z_365><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest prime divisor of $3^{15} + 11^9$?<|im_end|>
<|im_start|>assistant
<Z_169><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the smallest prime divisor of $3^{15} + 11^9$?<|im_end|>
<|im_start|>assistant
<Z_392><Z_467><Z_70><Z_374><Z_220><Z_313><Z_309><ANSWER>

Question: What is the smallest prime divisor of $3^{15} + 11^9$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many years was the money lent if the simple interest on rs. 4000 amounts to rs. 640 at an interest rate of 8%?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many years was the money lent if the simple interest on rs. 4000 amounts to rs. 640 at an interest rate of 8%?<|im_end|>
<|im_start|>assistant
<Z_477><Z_136><Z_483><Z_460><Z_255><Z_443><Z_185><Z_507><Z_274><Z_391><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many years was the money lent if the simple interest on rs. 4000 amounts to rs. 640 at an interest rate of 8%?<|im_end|>
<|im_start|>assistant
<Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
For how many years was the money lent if the simple interest on rs. 4000 amounts to rs. 640 at an interest rate of 8%?<|im_end|>
<|im_start|>assistant
<Z_340><Z_271><Z_465><Z_327><Z_67><Z_59><Z_344><Z_178><Z_81><Z_136><ANSWER>

Question: For how many years was the money lent if the simple interest on rs. 4000 amounts to rs. 640 at an interest rate of 8%?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of prime factors in the expression ( some number ) ^ 11 x ( 7 ) ^ 3 x ( 11 ) ^ 2. The total number of prime factors is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of prime factors in the expression ( some number ) ^ 11 x ( 7 ) ^ 3 x ( 11 ) ^ 2. The total number of prime factors is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_477><Z_382><Z_503><Z_350><Z_197><Z_227><Z_445><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of prime factors in the expression ( some number ) ^ 11 x ( 7 ) ^ 3 x ( 11 ) ^ 2. The total number of prime factors is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_496><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Find the total number of prime factors in the expression ( some number ) ^ 11 x ( 7 ) ^ 3 x ( 11 ) ^ 2. The total number of prime factors is 27. What is the number?<|im_end|>
<|im_start|>assistant
<Z_235><Z_455><Z_118><Z_386><Z_215><Z_355><Z_455><ANSWER>

Question: Find the total number of prime factors in the expression ( some number ) ^ 11 x ( 7 ) ^ 3 x ( 11 ) ^ 2. The total number of prime factors is 27. What is the number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_213><Z_321><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$.<|im_end|>
<|im_start|>assistant
<Z_109><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$.<|im_end|>
<|im_start|>assistant
<Z_501><Z_334><Z_448><ANSWER>

Question:
In $\triangle ABC$, $AC=5 \sqrt {2}$, $\cos C= \frac {3}{5}$, $B= \frac {\pi}{4}$.
(1) Find the length of $AB$;
(2) Find the area of $\triangle ABC$, denoted as $S_{\triangle ABC}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all of the solutions of the equation $\frac{6x}{30}=\frac{8}{x}$?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all of the solutions of the equation $\frac{6x}{30}=\frac{8}{x}$?<|im_end|>
<|im_start|>assistant
<Z_24><Z_371><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all of the solutions of the equation $\frac{6x}{30}=\frac{8}{x}$?<|im_end|>
<|im_start|>assistant
<Z_291><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of all of the solutions of the equation $\frac{6x}{30}=\frac{8}{x}$?<|im_end|>
<|im_start|>assistant
<Z_448><Z_477><ANSWER>

Question: What is the sum of all of the solutions of the equation $\frac{6x}{30}=\frac{8}{x}$?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train which is 300 m long is running at 80 kmph. It crosses a person moving at a certain speed in the same direction in 15 seconds. What is the speed of the person?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train which is 300 m long is running at 80 kmph. It crosses a person moving at a certain speed in the same direction in 15 seconds. What is the speed of the person?<|im_end|>
<|im_start|>assistant
<Z_369><Z_318><Z_276><Z_340><Z_115><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train which is 300 m long is running at 80 kmph. It crosses a person moving at a certain speed in the same direction in 15 seconds. What is the speed of the person?<|im_end|>
<|im_start|>assistant
<Z_446><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A train which is 300 m long is running at 80 kmph. It crosses a person moving at a certain speed in the same direction in 15 seconds. What is the speed of the person?<|im_end|>
<|im_start|>assistant
<Z_145><Z_67><Z_284><Z_484><Z_80><ANSWER>

Question: A train which is 300 m long is running at 80 kmph. It crosses a person moving at a certain speed in the same direction in 15 seconds. What is the speed of the person?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can five people line up in a straight line if the youngest person cannot be first or last in line?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can five people line up in a straight line if the youngest person cannot be first or last in line?<|im_end|>
<|im_start|>assistant
<Z_369><Z_362><Z_218><Z_194><Z_184><Z_264><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can five people line up in a straight line if the youngest person cannot be first or last in line?<|im_end|>
<|im_start|>assistant
<Z_188><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In how many ways can five people line up in a straight line if the youngest person cannot be first or last in line?<|im_end|>
<|im_start|>assistant
<Z_40><Z_456><Z_302><Z_180><Z_278><Z_483><ANSWER>

Question: In how many ways can five people line up in a straight line if the youngest person cannot be first or last in line?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cubical block of metal weighs 3 pounds and has a density of 'D' and is subject to an external force 'F'. How much will another cube of the same metal weigh if its sides are twice as long, the density of the metal is increased by 25%, and the external force applied on it is doubled?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cubical block of metal weighs 3 pounds and has a density of 'D' and is subject to an external force 'F'. How much will another cube of the same metal weigh if its sides are twice as long, the density of the metal is increased by 25%, and the external force applied on it is doubled?<|im_end|>
<|im_start|>assistant
<Z_369><Z_321><Z_28><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cubical block of metal weighs 3 pounds and has a density of 'D' and is subject to an external force 'F'. How much will another cube of the same metal weigh if its sides are twice as long, the density of the metal is increased by 25%, and the external force applied on it is doubled?<|im_end|>
<|im_start|>assistant
<Z_291><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A cubical block of metal weighs 3 pounds and has a density of 'D' and is subject to an external force 'F'. How much will another cube of the same metal weigh if its sides are twice as long, the density of the metal is increased by 25%, and the external force applied on it is doubled?<|im_end|>
<|im_start|>assistant
<Z_313><Z_299><Z_267><ANSWER>

Question: A cubical block of metal weighs 3 pounds and has a density of 'D' and is subject to an external force 'F'. How much will another cube of the same metal weigh if its sides are twice as long, the density of the metal is increased by 25%, and the external force applied on it is doubled?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.<|im_end|>
<|im_start|>assistant
<Z_24><Z_313><Z_478><Z_294><Z_349><Z_487><Z_101><Z_29><Z_111><Z_84><Z_449><Z_470><Z_64><Z_139><Z_309><Z_409><Z_213><Z_253><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.<|im_end|>
<|im_start|>assistant
<Z_342><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.<|im_end|>
<|im_start|>assistant
<Z_409><Z_15><Z_21><Z_404><Z_169><Z_332><Z_209><Z_12><Z_476><Z_324><Z_251><Z_187><Z_283><Z_380><Z_76><Z_366><Z_21><Z_487><ANSWER>

Question: The sum of the first 1500 terms of a geometric sequence is 300. The sum of the first 3000 terms is 570. Find the sum of the first 4500 terms.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom bought 14 boxes of chocolate candy, 10 boxes of fruit candy, and 8 boxes of caramel candy. He gave 8 chocolate boxes and 5 fruit boxes to his little brother. If each chocolate box has 3 pieces inside, each fruit box has 4 pieces, and each caramel box has 5 pieces, how many pieces of candy does Tom still have?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom bought 14 boxes of chocolate candy, 10 boxes of fruit candy, and 8 boxes of caramel candy. He gave 8 chocolate boxes and 5 fruit boxes to his little brother. If each chocolate box has 3 pieces inside, each fruit box has 4 pieces, and each caramel box has 5 pieces, how many pieces of candy does Tom still have?<|im_end|>
<|im_start|>assistant
<Z_477><Z_84><Z_169><Z_445><Z_169><Z_114><Z_60><Z_318><Z_381><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom bought 14 boxes of chocolate candy, 10 boxes of fruit candy, and 8 boxes of caramel candy. He gave 8 chocolate boxes and 5 fruit boxes to his little brother. If each chocolate box has 3 pieces inside, each fruit box has 4 pieces, and each caramel box has 5 pieces, how many pieces of candy does Tom still have?<|im_end|>
<|im_start|>assistant
<Z_15><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Tom bought 14 boxes of chocolate candy, 10 boxes of fruit candy, and 8 boxes of caramel candy. He gave 8 chocolate boxes and 5 fruit boxes to his little brother. If each chocolate box has 3 pieces inside, each fruit box has 4 pieces, and each caramel box has 5 pieces, how many pieces of candy does Tom still have?<|im_end|>
<|im_start|>assistant
<Z_37><Z_310><Z_234><Z_255><Z_223><Z_42><Z_472><Z_32><Z_133><ANSWER>

Question: Tom bought 14 boxes of chocolate candy, 10 boxes of fruit candy, and 8 boxes of caramel candy. He gave 8 chocolate boxes and 5 fruit boxes to his little brother. If each chocolate box has 3 pieces inside, each fruit box has 4 pieces, and each caramel box has 5 pieces, how many pieces of candy does Tom still have?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a\neq 0$, let set $A=\{x|x^{2}-x-6 < 0\}$, $B=\{x|x^{2}+2x-8\geqslant 0\}$, and $C=\{x|x^{2}-4ax+3a^{2} < 0\}$, and $C\subseteq(A\cap\neg_{R}B)$. Find the range of values for the real number $a$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a\neq 0$, let set $A=\{x|x^{2}-x-6 < 0\}$, $B=\{x|x^{2}+2x-8\geqslant 0\}$, and $C=\{x|x^{2}-4ax+3a^{2} < 0\}$, and $C\subseteq(A\cap\neg_{R}B)$. Find the range of values for the real number $a$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_426><Z_382><Z_55><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a\neq 0$, let set $A=\{x|x^{2}-x-6 < 0\}$, $B=\{x|x^{2}+2x-8\geqslant 0\}$, and $C=\{x|x^{2}-4ax+3a^{2} < 0\}$, and $C\subseteq(A\cap\neg_{R}B)$. Find the range of values for the real number $a$.<|im_end|>
<|im_start|>assistant
<Z_45><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that $a\neq 0$, let set $A=\{x|x^{2}-x-6 < 0\}$, $B=\{x|x^{2}+2x-8\geqslant 0\}$, and $C=\{x|x^{2}-4ax+3a^{2} < 0\}$, and $C\subseteq(A\cap\neg_{R}B)$. Find the range of values for the real number $a$.<|im_end|>
<|im_start|>assistant
<Z_371><Z_256><Z_337><Z_262><ANSWER>

Question: Given that $a\neq 0$, let set $A=\{x|x^{2}-x-6 < 0\}$, $B=\{x|x^{2}+2x-8\geqslant 0\}$, and $C=\{x|x^{2}-4ax+3a^{2} < 0\}$, and $C\subseteq(A\cap\neg_{R}B)$. Find the range of values for the real number $a$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $\{a_{n}\}$, if $a_{1}+a_{2}=4$ and $a_{10}+a_{9}=36$, find the value of $S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $\{a_{n}\}$, if $a_{1}+a_{2}=4$ and $a_{10}+a_{9}=36$, find the value of $S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_322><Z_483><Z_99><Z_60><Z_508><Z_415><Z_101><Z_387><Z_218><Z_364><Z_288><Z_26><Z_413><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $\{a_{n}\}$, if $a_{1}+a_{2}=4$ and $a_{10}+a_{9}=36$, find the value of $S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In the arithmetic sequence $\{a_{n}\}$, if $a_{1}+a_{2}=4$ and $a_{10}+a_{9}=36$, find the value of $S_{10}$.<|im_end|>
<|im_start|>assistant
<Z_158><Z_284><Z_225><Z_461><Z_410><Z_162><Z_394><Z_295><Z_313><Z_168><Z_153><Z_201><Z_191><Z_464><ANSWER>

Question: In the arithmetic sequence $\{a_{n}\}$, if $a_{1}+a_{2}=4$ and $a_{10}+a_{9}=36$, find the value of $S_{10}$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $S\_n$ represents the sum of its first $n$ terms, and $S\_{12} > 0$, $S\_{13} < 0$, find the smallest value of $n$ that makes $a\_n < 0$ true.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $S\_n$ represents the sum of its first $n$ terms, and $S\_{12} > 0$, $S\_{13} < 0$, find the smallest value of $n$ that makes $a\_n < 0$ true.<|im_end|>
<|im_start|>assistant
<Z_369><Z_306><Z_467><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $S\_n$ represents the sum of its first $n$ terms, and $S\_{12} > 0$, $S\_{13} < 0$, find the smallest value of $n$ that makes $a\_n < 0$ true.<|im_end|>
<|im_start|>assistant
<Z_461><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $S\_n$ represents the sum of its first $n$ terms, and $S\_{12} > 0$, $S\_{13} < 0$, find the smallest value of $n$ that makes $a\_n < 0$ true.<|im_end|>
<|im_start|>assistant
<Z_47><Z_151><Z_457><ANSWER>

Question: Given that the sequence $\{a\_n\}$ is an arithmetic sequence, $S\_n$ represents the sum of its first $n$ terms, and $S\_{12} > 0$, $S\_{13} < 0$, find the smallest value of $n$ that makes $a\_n < 0$ true.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = \frac{x + 8}{x}$. Define the sequence $(g_n)$ of functions by $g_1 = g$ and
\[ g_n = g \circ g_{n-1} \]
for all $n \geq 2$. For example,
\[ g_2(x) = g(g(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ g_3(x) = g(g_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 72}{9x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$. Find the number of elements in $T$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = \frac{x + 8}{x}$. Define the sequence $(g_n)$ of functions by $g_1 = g$ and
\[ g_n = g \circ g_{n-1} \]
for all $n \geq 2$. For example,
\[ g_2(x) = g(g(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ g_3(x) = g(g_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 72}{9x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$. Find the number of elements in $T$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_60><Z_453><Z_241><Z_185><Z_206><Z_331><Z_494><Z_71><Z_445><Z_378><Z_298><Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = \frac{x + 8}{x}$. Define the sequence $(g_n)$ of functions by $g_1 = g$ and
\[ g_n = g \circ g_{n-1} \]
for all $n \geq 2$. For example,
\[ g_2(x) = g(g(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ g_3(x) = g(g_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 72}{9x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$. Find the number of elements in $T$.<|im_end|>
<|im_start|>assistant
<Z_158><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Let $g(x) = \frac{x + 8}{x}$. Define the sequence $(g_n)$ of functions by $g_1 = g$ and
\[ g_n = g \circ g_{n-1} \]
for all $n \geq 2$. For example,
\[ g_2(x) = g(g(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ g_3(x) = g(g_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 72}{9x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$. Find the number of elements in $T$.<|im_end|>
<|im_start|>assistant
<Z_248><Z_310><Z_360><Z_141><Z_201><Z_356><Z_19><Z_179><Z_15><Z_284><Z_339><Z_438><Z_301><ANSWER>

Question:
Let $g(x) = \frac{x + 8}{x}$. Define the sequence $(g_n)$ of functions by $g_1 = g$ and
\[ g_n = g \circ g_{n-1} \]
for all $n \geq 2$. For example,
\[ g_2(x) = g(g(x)) = \frac{\frac{x + 8}{x} + 8}{\frac{x + 8}{x}} = \frac{9x + 8}{x + 8} \]
and
\[ g_3(x) = g(g_2(x)) = \frac{\frac{9x + 8}{x + 8} + 8}{\frac{9x + 8}{x + 8}} = \frac{17x + 72}{9x + 8}.\]
Let $T$ be the set of all real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$. Find the number of elements in $T$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The difference of two numbers is 1370. On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the larger number?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The difference of two numbers is 1370. On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the larger number?<|im_end|>
<|im_start|>assistant
<Z_477><Z_320><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The difference of two numbers is 1370. On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the larger number?<|im_end|>
<|im_start|>assistant
<Z_233><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
The difference of two numbers is 1370. On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the larger number?<|im_end|>
<|im_start|>assistant
<Z_508><Z_173><ANSWER>

Question: The difference of two numbers is 1370. On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the larger number?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subsidiary company of a certain group had a capital of 50 million yuan at the beginning of the year 2023. According to past experience, if all the capital is invested in production, the company's annual capital growth rate is 50%. The group now requires the company to remit 10 million yuan at the end of each year starting from 2023, and invest the remaining capital in production for the next year. Let $a_{n}$ be the remaining capital of the company after remitting the capital at the end of the $n$th year.

$(1)$ Find $a_{n}$;
$(2)$ If the remaining capital of the company at the end of the $m$th year exceeds 30 million yuan, find the minimum value of $m$.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subsidiary company of a certain group had a capital of 50 million yuan at the beginning of the year 2023. According to past experience, if all the capital is invested in production, the company's annual capital growth rate is 50%. The group now requires the company to remit 10 million yuan at the end of each year starting from 2023, and invest the remaining capital in production for the next year. Let $a_{n}$ be the remaining capital of the company after remitting the capital at the end of the $n$th year.

$(1)$ Find $a_{n}$;
$(2)$ If the remaining capital of the company at the end of the $m$th year exceeds 30 million yuan, find the minimum value of $m$.<|im_end|>
<|im_start|>assistant
<Z_477><Z_267><Z_450><Z_465><Z_400><Z_394><Z_178><Z_371><Z_330><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subsidiary company of a certain group had a capital of 50 million yuan at the beginning of the year 2023. According to past experience, if all the capital is invested in production, the company's annual capital growth rate is 50%. The group now requires the company to remit 10 million yuan at the end of each year starting from 2023, and invest the remaining capital in production for the next year. Let $a_{n}$ be the remaining capital of the company after remitting the capital at the end of the $n$th year.

$(1)$ Find $a_{n}$;
$(2)$ If the remaining capital of the company at the end of the $m$th year exceeds 30 million yuan, find the minimum value of $m$.<|im_end|>
<|im_start|>assistant
<Z_218><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A subsidiary company of a certain group had a capital of 50 million yuan at the beginning of the year 2023. According to past experience, if all the capital is invested in production, the company's annual capital growth rate is 50%. The group now requires the company to remit 10 million yuan at the end of each year starting from 2023, and invest the remaining capital in production for the next year. Let $a_{n}$ be the remaining capital of the company after remitting the capital at the end of the $n$th year.

$(1)$ Find $a_{n}$;
$(2)$ If the remaining capital of the company at the end of the $m$th year exceeds 30 million yuan, find the minimum value of $m$.<|im_end|>
<|im_start|>assistant
<Z_195><Z_334><Z_179><Z_21><Z_396><Z_18><Z_241><Z_73><Z_372><ANSWER>

Question:
A subsidiary company of a certain group had a capital of 50 million yuan at the beginning of the year 2023. According to past experience, if all the capital is invested in production, the company's annual capital growth rate is 50%. The group now requires the company to remit 10 million yuan at the end of each year starting from 2023, and invest the remaining capital in production for the next year. Let $a_{n}$ be the remaining capital of the company after remitting the capital at the end of the $n$th year.

$(1)$ Find $a_{n}$;
$(2)$ If the remaining capital of the company at the end of the $m$th year exceeds 30 million yuan, find the minimum value of $m$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $f(x)$ defined on $\mathbb{R}$ is even and satisfies $f(x+4)=f(x)$. When $x\in[0,2]$, $f(x)=3^{x}$. Determine the value of $f(-9)$.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $f(x)$ defined on $\mathbb{R}$ is even and satisfies $f(x+4)=f(x)$. When $x\in[0,2]$, $f(x)=3^{x}$. Determine the value of $f(-9)$.<|im_end|>
<|im_start|>assistant
<Z_24><Z_307><Z_393><Z_185><Z_246><Z_439><Z_313><Z_240><Z_425><Z_296><Z_77><Z_245><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $f(x)$ defined on $\mathbb{R}$ is even and satisfies $f(x+4)=f(x)$. When $x\in[0,2]$, $f(x)=3^{x}$. Determine the value of $f(-9)$.<|im_end|>
<|im_start|>assistant
<Z_317><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A function $f(x)$ defined on $\mathbb{R}$ is even and satisfies $f(x+4)=f(x)$. When $x\in[0,2]$, $f(x)=3^{x}$. Determine the value of $f(-9)$.<|im_end|>
<|im_start|>assistant
<Z_465><Z_22><Z_366><Z_377><Z_322><Z_348><Z_340><Z_128><Z_410><Z_52><Z_147><Z_268><ANSWER>

Question: A function $f(x)$ defined on $\mathbb{R}$ is even and satisfies $f(x+4)=f(x)$. When $x\in[0,2]$, $f(x)=3^{x}$. Determine the value of $f(-9)$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A natural number \( n \) is called interesting if \( 2n \) is a perfect square and \( 15n \) is a perfect cube. Find the smallest interesting number.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A natural number \( n \) is called interesting if \( 2n \) is a perfect square and \( 15n \) is a perfect cube. Find the smallest interesting number.<|im_end|>
<|im_start|>assistant
<Z_477><Z_77><Z_217><Z_366><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A natural number \( n \) is called interesting if \( 2n \) is a perfect square and \( 15n \) is a perfect cube. Find the smallest interesting number.<|im_end|>
<|im_start|>assistant
<Z_432><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A natural number \( n \) is called interesting if \( 2n \) is a perfect square and \( 15n \) is a perfect cube. Find the smallest interesting number.<|im_end|>
<|im_start|>assistant
<Z_281><Z_122><Z_419><Z_414><ANSWER>

Question: A natural number \( n \) is called interesting if \( 2n \) is a perfect square and \( 15n \) is a perfect cube. Find the smallest interesting number.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A segment \(P Q\) is divided into several smaller segments. A square is constructed on each of these segments (see image).

What is the length of the path along the arrows if the length of the segment \(P Q\) is 73? If necessary, round the answer to 0.01 or write the answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A segment \(P Q\) is divided into several smaller segments. A square is constructed on each of these segments (see image).

What is the length of the path along the arrows if the length of the segment \(P Q\) is 73? If necessary, round the answer to 0.01 or write the answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_24><Z_469><Z_274><Z_217><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A segment \(P Q\) is divided into several smaller segments. A square is constructed on each of these segments (see image).

What is the length of the path along the arrows if the length of the segment \(P Q\) is 73? If necessary, round the answer to 0.01 or write the answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_424><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A segment \(P Q\) is divided into several smaller segments. A square is constructed on each of these segments (see image).

What is the length of the path along the arrows if the length of the segment \(P Q\) is 73? If necessary, round the answer to 0.01 or write the answer as a common fraction.<|im_end|>
<|im_start|>assistant
<Z_464><Z_349><Z_323><Z_309><ANSWER>

Question:
A segment \(P Q\) is divided into several smaller segments. A square is constructed on each of these segments (see image).

What is the length of the path along the arrows if the length of the segment \(P Q\) is 73? If necessary, round the answer to 0.01 or write the answer as a common fraction.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?<|im_end|>
<|im_start|>assistant
<Z_24><Z_488><Z_157><Z_433><Z_465><Z_497><Z_202><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?<|im_end|>
<|im_start|>assistant
<Z_180><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?<|im_end|>
<|im_start|>assistant
<Z_207><Z_329><Z_475><Z_143><Z_151><Z_188><Z_485><ANSWER>

Question: In a regular 1000-gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the chosen diagonals, at least two have the same length?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Karolína wrote all three-digit numbers composed of the digits 1, 2, and 3, where no digit repeats, and 2 was in the tens place. Nikola wrote all three-digit numbers composed of the digits 4, 5, and 6, where no digit repeats. Kuba selected one number from Karolína and one number from Nikola such that the sum of these two numbers was even.

What was the digit in the units place of the product of the numbers selected by Kuba? Determine all possibilities.<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Karolína wrote all three-digit numbers composed of the digits 1, 2, and 3, where no digit repeats, and 2 was in the tens place. Nikola wrote all three-digit numbers composed of the digits 4, 5, and 6, where no digit repeats. Kuba selected one number from Karolína and one number from Nikola such that the sum of these two numbers was even.

What was the digit in the units place of the product of the numbers selected by Kuba? Determine all possibilities.<|im_end|>
<|im_start|>assistant
<Z_24><Z_503><Z_467><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Karolína wrote all three-digit numbers composed of the digits 1, 2, and 3, where no digit repeats, and 2 was in the tens place. Nikola wrote all three-digit numbers composed of the digits 4, 5, and 6, where no digit repeats. Kuba selected one number from Karolína and one number from Nikola such that the sum of these two numbers was even.

What was the digit in the units place of the product of the numbers selected by Kuba? Determine all possibilities.<|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
Karolína wrote all three-digit numbers composed of the digits 1, 2, and 3, where no digit repeats, and 2 was in the tens place. Nikola wrote all three-digit numbers composed of the digits 4, 5, and 6, where no digit repeats. Kuba selected one number from Karolína and one number from Nikola such that the sum of these two numbers was even.

What was the digit in the units place of the product of the numbers selected by Kuba? Determine all possibilities.<|im_end|>
<|im_start|>assistant
<Z_95><Z_111><Z_16><ANSWER>

Question:
Karolína wrote all three-digit numbers composed of the digits 1, 2, and 3, where no digit repeats, and 2 was in the tens place. Nikola wrote all three-digit numbers composed of the digits 4, 5, and 6, where no digit repeats. Kuba selected one number from Karolína and one number from Nikola such that the sum of these two numbers was even.

What was the digit in the units place of the product of the numbers selected by Kuba? Determine all possibilities.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the number of positive integer solutions for the equation x + y + z + t = 10 is<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the number of positive integer solutions for the equation x + y + z + t = 10 is<|im_end|>
<|im_start|>assistant
<Z_24><Z_469><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the number of positive integer solutions for the equation x + y + z + t = 10 is<|im_end|>
<|im_start|>assistant
<Z_29><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
the number of positive integer solutions for the equation x + y + z + t = 10 is<|im_end|>
<|im_start|>assistant
<Z_132><Z_243><ANSWER>

Question: the number of positive integer solutions for the equation x + y + z + t = 10 is


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Granger has five times as many spots as his brother, Cisco. Cisco has 5 less than a certain fraction of his cousin, Rover's spots. Rover has 46 spots. Granger and Cisco have combined 108 spots. What is the fraction of Rover's spots that Cisco has 5 less than?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Granger has five times as many spots as his brother, Cisco. Cisco has 5 less than a certain fraction of his cousin, Rover's spots. Rover has 46 spots. Granger and Cisco have combined 108 spots. What is the fraction of Rover's spots that Cisco has 5 less than?<|im_end|>
<|im_start|>assistant
<Z_24><Z_278><Z_487><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Granger has five times as many spots as his brother, Cisco. Cisco has 5 less than a certain fraction of his cousin, Rover's spots. Rover has 46 spots. Granger and Cisco have combined 108 spots. What is the fraction of Rover's spots that Cisco has 5 less than?<|im_end|>
<|im_start|>assistant
<Z_376><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Granger has five times as many spots as his brother, Cisco. Cisco has 5 less than a certain fraction of his cousin, Rover's spots. Rover has 46 spots. Granger and Cisco have combined 108 spots. What is the fraction of Rover's spots that Cisco has 5 less than?<|im_end|>
<|im_start|>assistant
<Z_29><Z_507><Z_457><ANSWER>

Question: Granger has five times as many spots as his brother, Cisco. Cisco has 5 less than a certain fraction of his cousin, Rover's spots. Rover has 46 spots. Granger and Cisco have combined 108 spots. What is the fraction of Rover's spots that Cisco has 5 less than?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were a total of 8 football games this year, 4 are played at night. Keith missed some of the games. Keith went to 4 football games in total. How many football games did Keith miss?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were a total of 8 football games this year, 4 are played at night. Keith missed some of the games. Keith went to 4 football games in total. How many football games did Keith miss?<|im_end|>
<|im_start|>assistant
<Z_24><Z_210><Z_288><Z_171><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were a total of 8 football games this year, 4 are played at night. Keith missed some of the games. Keith went to 4 football games in total. How many football games did Keith miss?<|im_end|>
<|im_start|>assistant
<Z_302><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There were a total of 8 football games this year, 4 are played at night. Keith missed some of the games. Keith went to 4 football games in total. How many football games did Keith miss?<|im_end|>
<|im_start|>assistant
<Z_431><Z_458><Z_428><Z_473><ANSWER>

Question: There were a total of 8 football games this year, 4 are played at night. Keith missed some of the games. Keith went to 4 football games in total. How many football games did Keith miss?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use the Horner's Rule to calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) at \\(x=-4\\). The value of \\(V_{4}\\) is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use the Horner's Rule to calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) at \\(x=-4\\). The value of \\(V_{4}\\) is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_24><Z_205><Z_58><Z_509><Z_334><Z_494><Z_79><Z_478><Z_135><Z_318><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use the Horner's Rule to calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) at \\(x=-4\\). The value of \\(V_{4}\\) is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_418><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Use the Horner's Rule to calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) at \\(x=-4\\). The value of \\(V_{4}\\) is \_\_\_\_\_\_.<|im_end|>
<|im_start|>assistant
<Z_476><Z_232><Z_241><Z_461><Z_23><Z_511><Z_388><Z_62><Z_364><Z_132><ANSWER>

Question: Use the Horner's Rule to calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) at \\(x=-4\\). The value of \\(V_{4}\\) is \_\_\_\_\_\_.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?<|im_end|>
<|im_start|>assistant
<Z_369><Z_306><Z_279><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?<|im_end|>
<|im_start|>assistant
<Z_452><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?<|im_end|>
<|im_start|>assistant
<Z_447><Z_422><Z_52><ANSWER>

Question: Of the numbers 1, 2, 3, ..., 20, which number less than 20 has the greatest number of divisors and is also a composite number?


Question:
Compute
\[\frac{(12^4 + 324)(24^4 + 324)(36^4 + 324)(48^4 + 324)(60^4 + 324)}{(6^4 + 324)(18^4 + 324)(30^4 + 324)(42^4 + 324)(54^4 + 324)}.\]


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Having scored 95 runs in the 19th inning, a cricketer increases his average score by a certain amount. His average score after 19 innings is 23. By how much did his average score increase?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Having scored 95 runs in the 19th inning, a cricketer increases his average score by a certain amount. His average score after 19 innings is 23. By how much did his average score increase?<|im_end|>
<|im_start|>assistant
<Z_477><Z_250><Z_351><Z_430><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Having scored 95 runs in the 19th inning, a cricketer increases his average score by a certain amount. His average score after 19 innings is 23. By how much did his average score increase?<|im_end|>
<|im_start|>assistant
<Z_508><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Having scored 95 runs in the 19th inning, a cricketer increases his average score by a certain amount. His average score after 19 innings is 23. By how much did his average score increase?<|im_end|>
<|im_start|>assistant
<Z_426><Z_355><Z_151><Z_8><ANSWER>

Question: Having scored 95 runs in the 19th inning, a cricketer increases his average score by a certain amount. His average score after 19 innings is 23. By how much did his average score increase?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a $70 \times 70$ grid, 2018 cells were cut out. Prove that the grid breaks into no more than 2018 pieces. Two pieces that share only vertex points are considered disconnected.<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a $70 \times 70$ grid, 2018 cells were cut out. Prove that the grid breaks into no more than 2018 pieces. Two pieces that share only vertex points are considered disconnected.<|im_end|>
<|im_start|>assistant
<Z_24><Z_503><Z_87><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a $70 \times 70$ grid, 2018 cells were cut out. Prove that the grid breaks into no more than 2018 pieces. Two pieces that share only vertex points are considered disconnected.<|im_end|>
<|im_start|>assistant
<Z_455><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From a $70 \times 70$ grid, 2018 cells were cut out. Prove that the grid breaks into no more than 2018 pieces. Two pieces that share only vertex points are considered disconnected.<|im_end|>
<|im_start|>assistant
<Z_359><Z_7><Z_407><ANSWER>

Question: From a $70 \times 70$ grid, 2018 cells were cut out. Prove that the grid breaks into no more than 2018 pieces. Two pieces that share only vertex points are considered disconnected.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
find the principle on a certain sum of money at 9 % per annum for 2 2 / 5 years if the amount being rs . 1120 ?<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
find the principle on a certain sum of money at 9 % per annum for 2 2 / 5 years if the amount being rs . 1120 ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_418><Z_135><Z_209><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
find the principle on a certain sum of money at 9 % per annum for 2 2 / 5 years if the amount being rs . 1120 ?<|im_end|>
<|im_start|>assistant
<Z_510><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
find the principle on a certain sum of money at 9 % per annum for 2 2 / 5 years if the amount being rs . 1120 ?<|im_end|>
<|im_start|>assistant
<Z_508><Z_127><Z_262><Z_382><ANSWER>

Question: find the principle on a certain sum of money at 9 % per annum for 2 2 / 5 years if the amount being rs . 1120 ?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie grew 139 turnips . Benny grew 113 turnips. They grew _____ turnips in all .<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie grew 139 turnips . Benny grew 113 turnips. They grew _____ turnips in all .<|im_end|>
<|im_start|>assistant
<Z_477><Z_404><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie grew 139 turnips . Benny grew 113 turnips. They grew _____ turnips in all .<|im_end|>
<|im_start|>assistant
<Z_254><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Melanie grew 139 turnips . Benny grew 113 turnips. They grew _____ turnips in all .<|im_end|>
<|im_start|>assistant
<Z_313><Z_500><ANSWER>

Question: Melanie grew 139 turnips . Benny grew 113 turnips. They grew _____ turnips in all .


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 160 tissues inside a tissue box.   If Tucker bought 3 boxes, and used 210 tissues while he was sick with the flu, how many tissues would he have left?<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 160 tissues inside a tissue box.   If Tucker bought 3 boxes, and used 210 tissues while he was sick with the flu, how many tissues would he have left?<|im_end|>
<|im_start|>assistant
<Z_369><Z_320><Z_439><Z_401><Z_474><Z_219><Z_315><Z_229><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 160 tissues inside a tissue box.   If Tucker bought 3 boxes, and used 210 tissues while he was sick with the flu, how many tissues would he have left?<|im_end|>
<|im_start|>assistant
<Z_65><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
There are 160 tissues inside a tissue box.   If Tucker bought 3 boxes, and used 210 tissues while he was sick with the flu, how many tissues would he have left?<|im_end|>
<|im_start|>assistant
<Z_454><Z_183><Z_299><Z_214><Z_21><Z_478><Z_189><Z_131><ANSWER>

Question: There are 160 tissues inside a tissue box.   If Tucker bought 3 boxes, and used 210 tissues while he was sick with the flu, how many tissues would he have left?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)<|im_end|>
<|im_start|>assistant
<Z_24><Z_154><Z_398><Z_501><Z_137><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)<|im_end|>
<|im_start|>assistant
<Z_103><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)<|im_end|>
<|im_start|>assistant
<Z_329><Z_330><Z_167><Z_223><Z_177><ANSWER>

Question: Solve the congruence $13n \equiv 9 \pmod{47}$, as a residue modulo 47. (Provide an answer between 0 and 46.)


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will necessarily be two numbers whose difference is greater than 672 and less than 1344?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will necessarily be two numbers whose difference is greater than 672 and less than 1344?<|im_end|>
<|im_start|>assistant
<Z_369><Z_125><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will necessarily be two numbers whose difference is greater than 672 and less than 1344?<|im_end|>
<|im_start|>assistant
<Z_23><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will necessarily be two numbers whose difference is greater than 672 and less than 1344?<|im_end|>
<|im_start|>assistant
<Z_152><Z_307><ANSWER>

Question: From the numbers 1, 2, 3, ..., 2016, $k$ numbers are chosen. What is the smallest $k$ such that among the chosen numbers, there will necessarily be two numbers whose difference is greater than 672 and less than 1344?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular box has width $10$ inches, length $20$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $40$ square inches. Determine $m+n$.<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular box has width $10$ inches, length $20$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $40$ square inches. Determine $m+n$.<|im_end|>
<|im_start|>assistant
<Z_369><Z_48><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular box has width $10$ inches, length $20$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $40$ square inches. Determine $m+n$.<|im_end|>
<|im_start|>assistant
<Z_283><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A rectangular box has width $10$ inches, length $20$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $40$ square inches. Determine $m+n$.<|im_end|>
<|im_start|>assistant
<Z_266><Z_150><ANSWER>

Question: A rectangular box has width $10$ inches, length $20$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $40$ square inches. Determine $m+n$.


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A shopkeeper sells 20% of his stock at 10% profit and sells the remaining at a loss of 5%. He incurred a certain overall loss amount. The total worth of the stock is Rs. 19,999.99. What is the overall loss amount?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A shopkeeper sells 20% of his stock at 10% profit and sells the remaining at a loss of 5%. He incurred a certain overall loss amount. The total worth of the stock is Rs. 19,999.99. What is the overall loss amount?<|im_end|>
<|im_start|>assistant
<Z_477><Z_155><Z_507><Z_106><Z_75><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A shopkeeper sells 20% of his stock at 10% profit and sells the remaining at a loss of 5%. He incurred a certain overall loss amount. The total worth of the stock is Rs. 19,999.99. What is the overall loss amount?<|im_end|>
<|im_start|>assistant
<Z_234><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
A shopkeeper sells 20% of his stock at 10% profit and sells the remaining at a loss of 5%. He incurred a certain overall loss amount. The total worth of the stock is Rs. 19,999.99. What is the overall loss amount?<|im_end|>
<|im_start|>assistant
<Z_475><Z_279><Z_97><Z_49><Z_191><ANSWER>

Question: A shopkeeper sells 20% of his stock at 10% profit and sells the remaining at a loss of 5%. He incurred a certain overall loss amount. The total worth of the stock is Rs. 19,999.99. What is the overall loss amount?


<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jeff has 16 GB of storage on his phone. He is already using 4 GB. If a song takes up about 30MB of storage, how many songs can Jeff store on his phone? (There are 1000 MB in a GB).<|im_end|>
<|im_start|>assistant
<Z_477><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jeff has 16 GB of storage on his phone. He is already using 4 GB. If a song takes up about 30MB of storage, how many songs can Jeff store on his phone? (There are 1000 MB in a GB).<|im_end|>
<|im_start|>assistant
<Z_369><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jeff has 16 GB of storage on his phone. He is already using 4 GB. If a song takes up about 30MB of storage, how many songs can Jeff store on his phone? (There are 1000 MB in a GB).<|im_end|>
<|im_start|>assistant
<Z_272><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
Jeff has 16 GB of storage on his phone. He is already using 4 GB. If a song takes up about 30MB of storage, how many songs can Jeff store on his phone? (There are 1000 MB in a GB).<|im_end|>
<|im_start|>assistant
<Z_94><ANSWER>

Question: Jeff has 16 GB of storage on his phone. He is already using 4 GB. If a song takes up about 30MB of storage, how many songs can Jeff store on his phone? (There are 1000 MB in a GB).


<|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 digits of the integer value of ( 8 × 10 ^ {( 2 ^ 8 )} ) × ( 10 × 10 ^ {log_2( 2 ^ 12 )} ) ?<|im_end|>
<|im_start|>assistant
<Z_24><ANSWER>

<|im_start|>system
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>user
What is the sum of digits of the integer value of ( 8 × 10 ^ {( 2 ^ 8 )} ) × ( 10 × 10 ^ {log_2( 2 ^ 12 )} ) ?<|im_end|>
<|im_start|>assistant
<Z_477><Z_382><Z_143><Z_343><Z_118><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 digits of the integer value of ( 8 × 10 ^ {( 2 ^ 8 )} ) × ( 10 × 10 ^ {log_2( 2 ^ 12 )} ) ?<|im_end|>
<|im_start|>assistant
<Z_91><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 digits of the integer value of ( 8 × 10 ^ {( 2 ^ 8 )} ) × ( 10 × 10 ^ {log_2( 2 ^ 12 )} ) ?<|im_end|>
<|im_start|>assistant
<Z_324><Z_397><Z_282><Z_134><Z_194><ANSWER>

Question: What is the sum of digits of the integer value of ( 8 × 10 ^ {( 2 ^ 8 )} ) × ( 10 × 10 ^ {log_2( 2 ^ 12 )} ) ?