
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  $P(x,y)$  be a polynomial such that  $\deg_x(P), \deg_y(P)\le 2020$  and \[P(i,j)=\binom{i+j}{i}\] over all  $2021^2$  ordered pairs  $(i,j)$  with  $0\leq i,j\leq 2020$ . Find the remainder when  $P(4040, 4040)$  is divided by  $2017$ .

Note:  $\deg_x (P)$  is the highest exponent of  $x$  in a nonzero term of  $P(x,y)$ .  $\deg_y (P)$  is defined similarly.

*Proposed by Michael Ren*

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