
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

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Let \( M \) be a point in the interior of triangle \( ABC \). Let \( A' \) lie on \( BC \) such that \( MA' \) is perpendicular to \( BC \). Similarly, define point \( B' \) on \( CA \) and point \( C' \) on \( AB \) such that \( MB' \) is perpendicular to \( CA \) and \( MC' \) is perpendicular to \( AB \) respectively. Define the function
\[ p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}. \]
Determine, with proof, the location of \( M \) such that \( p(M) \) is maximal. Let \( \mu(ABC) \) denote this maximum value. For which triangles \( ABC \) is the value of \( \mu(ABC) \) maximal?

Present the answer in LaTex format: \boxed{Your answer}
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