<EXAMPLE 1>
<Question>
A palindrome is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Therefore, 01:10 may not be used.)
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this specific problem—a relatively straightforward combinatorial counting question on a digital clock where exhaustive checking is both feasible and error minimizing—the Program-Aided Language (PAL) approach stands out. It directly leverages a computer to enumerate and verify all possible times conforming to the unique formatting constraints while avoiding an overexposure of internal reasoning steps.\n\nSelected method: PAL (Program-Aided Language, Coding Only)
</Reasoning>
<Answer>
Selected method: PAL.
</Answer>
</EXAMPLE 1>

<EXAMPLE 2>
<Question>
A piece of cheese is located at $(12,10)$ in a coordinate plane.  A mouse is at $(4,-2)$ and is running  up  the line $y=-5x+18$.  At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$?
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this specific problem—a geometry problem involving distances and perpendicular intersections—the Chain-of-Thought method is likely the more effective approach. It promotes understanding of the underlying geometric reasoning and highlights the process behind finding the point where the mouse’s distance from the cheese is minimized.\n\nSelected method: COT (Chain of Thought, Language Only)
</Reasoning>
<Answer>
Selected method: COT.
</Answer>
</EXAMPLE 2>

<EXAMPLE 3>
<Question>
Evaluate $(2-w)(2-w^2)\cdots(2-w^{10})$ where $w=e^{2\pi i/11}.$
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this specific question, since the underlying algebraic structure (via cyclotomic polynomials) offers an elegant, exact answer, the chain-of-thought method is more effective and insightful. \n\nSelected method: COT (Chain of Thought, Language Only).
</Reasoning>
<Answer>
Selected method: COT.
</Answer>
</EXAMPLE 3>

<EXAMPLE 4>
<Question>
Four standard, six-sided dice are to be rolled. If the product of their values turns out to be an even number, what is the probability their sum is odd? Express your answer as a common fraction.
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this specific question, because the number of outcomes is small and the conditions can be programmed explicitly, the program-aided approach minimizes error risk and offers a quick, systematic solution.\n\nSelected method: PAL (Program-Aided Language, Coding Only)
</Reasoning>
<Answer>
Selected method: PAL
</Answer>
</EXAMPLE 4>

<EXAMPLE 5>
<Question>
In trapezoid $ABCD$ the lengths of the bases $AB$ and $CD$ are 8 and 17 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this trapezoid geometry problem, the chain‐of‐thought (COT) method allows you to gradually set up the problem by assigning variables, drawing relationships (such as the similarity of triangles), and then carefully deriving the necessary formulas. This process can enhance understanding and make it easier to check each individual step, especially when dealing with geometric properties and ratios.
On the other hand, the program‐aided language (PAL) method would involve writing a Python program to symbolically manipulate or numerically approach the problem. While Python can be very powerful, setting up the symbolic algebra needed for a clear derivation of an algebraic ratio may require additional packages or careful coding. This approach might risk making the solution less transparent and could introduce potential programming errors for a problem that is more elegantly handled through traditional reasoning.
Given that this problem benefits from step‐by‐step geometric reasoning and clear algebraic manipulation, the chain‐of‐thought method is more effective and straightforward for this specific question.\n\nSelected method: COT (Chain of Thought, Language Only).
</Reasoning>
<Answer>
Selected method: COT.
</Answer>
</EXAMPLE 5>

<EXAMPLE 6>
<Question>
A 4-inch by 6-inch picture is enlarged for framing  by tripling its dimensions.  A 2-inch-wide border  is then placed around each side of the enlarged  picture, as shown.  Thin metal framing is sold only  in increments of one foot.  What is the minimum  number of linear feet of framing that must be  purchased to go around the perimeter of the border?

[asy]

draw((0,0)--(14,0)--(14,20)--(0,20)--cycle,linewidth(2));

draw((4,4)--(10,4)--(10,16)--(4,16)--cycle);

label("border",(7,17),N);

label("picture",(7,8),N);

label("frame",(14,5),E);

draw((17.5,7.5)--(14.5,7.5),Arrow);
draw((10.5,7.5)--(13.5,7.5),Arrow);

[/asy]
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For THIS SPECIFIC QUESTION, the arithmetic and geometry are simple and direct. It is crucial to make sure that each conceptual step (enlargement, addition of borders, perimeter computation, and conversion unit) is clearly understood and communicated. The chain-of-thought method naturally lends itself to this clarity without the overhead of translating the problem into code. While coding could give the answer reliably, it may hide the reasoning steps that are important to verify each part of the computation.\n\nSelected method: COT.
</Reasoning>
<Answer>
Selected method: COT.
</Answer>
</EXAMPLE 6>

<EXAMPLE 7>
<Question>
Find the minimum of the function \[\frac{xy}{x^2 + y^2}\] in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this specific question—finding the minimum of a rational function over a small, compact domain—the program-aided approach is more directly effective. It reduces the intellectual overhead of dealing with all the subtle boundary and interior cases manually and tends to be more reliable given the tight constraints./n/nSelected method: PAL.
</Reasoning>
<Answer>
Selected method: PAL.
</Answer>
</EXAMPLE 7>

<EXAMPLE 8>
<Question>
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
</Question>
<Options>
COT (Chain of Thought, Language Only) – Solve the problem by reasoning through each step in natural language, breaking it down logically.
PAL (Program-Aided Language, Coding Only) – Directly write a Python program to solve the problem.
</Options>
<Reasoning>
For this specific question, where the answer hinges on a precise numerical property and where direct computational verification seems feasible (once the search boundaries are known or estimated), the Program-Aided Language (PAL) approach is more effective. It allows for a systematic exploration of candidates with the precision that the problem demands without running into lengthy manual derivations./n/nSelected method: PAL.
</Reasoning>
<Answer>
Selected method: PAL.
</Answer>
</EXAMPLE 8>

The following is the problem you need to solve.

