
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
We call polynomials  $A(x) = a_n x^n +. . .+a_1 x+a_0$  and  $B(x) = b_m x^m +. . .+b_1 x+b_0$ ( $a_n b_m \neq 0$ ) similar if the following conditions hold: $(i)$   $n = m$ ; $(ii)$  There is a permutation  $\pi$  of the set  $\{ 0, 1, . . . , n\} $  such that  $b_i = a_{\pi (i)}$  for each  $i \in {0, 1, . . . , n}$ .
Let  $P(x)$  and  $Q(x)$  be similar polynomials with integer coefficients. Given that $P(16) = 3^{2012}$ , find the smallest possible value of  $|Q(3^{2012})|$ .

*Proposed by Milos Milosavljevic*

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