
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 that  $P(x)$  is a monic quadratic polynomial satisfying  $aP(a) = 20P(20) = 22P(22)$  for some integer  $a\neq 20, 22$ . Find the minimum possible positive value of  $P(0)$ .

*Proposed by Andrew Wu*

(Note: wording changed from original to specify that  $a \neq 20, 22$ .)

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