
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
For each integer  $1\le j\le 2017$ , let  $S_j$  denote the set of integers  $0\le i\le 2^{2017} - 1$  such that  $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$  is an odd integer. Let  $P$  be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right)\in\{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by  $2017$ .

*Proposed by Ashwin Sah*

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