
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
Suppose  $P(x)$  is a monic polynomial of degree  $2023$  such that  $P(k) = k^{2023}P(1-\frac{1}{k})$  for every positive integer  $1 \leq k \leq 2023$ . Then  $P(-1) = \frac{a}{b}$  where  $a$  and  $b$  are relatively prime integers. Compute the unique integer  $0 \leq n < 2027$  such that  $bn-a$  is divisible by the prime  $2027$ .

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