
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
Let  $N$  be the number of polynomials  $P(x_1, x_2, \dots, x_{2016})$  of degree at most  $2015$  with coefficients in the set  $\{0, 1, 2 \}$  such that  $P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3}$  for all  $(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.$ Compute the remainder when  $v_3(N)$  is divided by  $2011$ , where  $v_3(N)$  denotes the largest integer  $k$  such that  $3^k | N.$ *Proposed by Yang Liu*

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