
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
Find the smallest possible value of a real number  $c$  such that for any  $2012$ -degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients,  we can obtain a new polynomial  $Q(x)$  by multiplying some of its coefficients by  $-1$  such that every root  $z$  of  $Q(x)$  satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

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