
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

...

Next action: [NEXT ACTION NAME]


system
For \( n \geqslant 1 \), let \( S_n \) be the set of polynomials of the form \( p(z) = z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + 1 \), where \( a_1, a_2, \cdots, a_{n-1} \) vary over all complex numbers. Find the value of \( M_n = \min_{p \in S_n} \left( \max_{|z| = 1} |p(z)| \right) \).

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