
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
A sequence of polynomials is defined by the recursion  $P_1(x) = x+1$  and  $$ P_{n}(x) = \frac{(P_{n-1}(x)+1)^5 - (P_{n-1}(-x)+1)^5}{2} $$  for all  $n \geq 2$ . Find the remainder when  $P_{2022}(1)$  is divided by  $1000$ .

*Proposed by **treemath***

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