
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  $S$  be the set of all polynomials  $Q(x,y,z)$  with coefficients in  $\{0,1\}$  such that there exists a homogeneous polynomial  $P(x,y,z)$  of degree  $2016$  with integer coefficients and a polynomial  $R(x,y,z)$  with integer coefficients so that \[P(x,y,z) Q(x,y,z) = P(yz,zx,xy)+2R(x,y,z)\] and  $P(1,1,1)$  is odd. Determine the size of  $S$ .

Note: A homogeneous polynomial of degree  $d$  consists solely of terms of degree  $d$ .

*Proposed by Vincent Huang*

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