
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
A *mirrored polynomial* is a polynomial  $f$  of degree  $100$  with real coefficients such that the  $x^{50}$  coefficient of  $f$  is  $1$ , and  $f(x) = x^{100} f(1/x)$  holds for all real nonzero  $x$ .  Find the smallest real constant  $C$  such that any mirrored polynomial  $f$  satisfying  $f(1) \ge C$  has a complex root  $z$  obeying  $|z| = 1$ .

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