<EXAMPLE 1>
<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>
Below is a brief assessment of each proposed method with respect to solving this specific question (calculating the probability that at least 15 cents come up heads when flipping four distinct coins):
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.
</Reasoning>
<Answer>
Selected method: PAL.
</Answer>
</EXAMPLE 1>

<EXAMPLE 2>
<Question>
What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A( -2, 0)$ and $B(0,4)$?
</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>
Because this is a single, concise geometry question—finding a point on the \( x \)-axis that is equidistant from two points—a straightforward algebraic or geometric approach is typically quick and clear. A step-by-step, purely language-based explanation (COT) is perfectly sufficient and easily handled without requiring the support of code. Thus, for this specific problem, COT (Chain of Thought, Language Only) is the most effective method.
</Reasoning>
<Answer>
Selected method: COT.
</Answer>
</EXAMPLE 2>

<EXAMPLE 3>
<Question>
    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 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.
</Reasoning>
<Answer>
Selected method: PAL
</Answer>
</EXAMPLE 3>

<EXAMPLE 4>
<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.
</Reasoning>
<Answer>
Selected method: COT
</Answer>
</EXAMPLE 4>

The following is the problem you need to solve.

<Question>
{Premises}
<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>
