
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  $P(x) = x^3 - 3x^2 + 3$ . For how many positive integers  $n < 1000$  does there not exist a pair  $(a, b)$  of positive integers such that the equation
\[ \underbrace{P(P(\dots P}_{a \text{ times}}(x)\dots))=\underbrace{P(P(\dots P}_{b \text{ times}}(x)\dots))\]
has exactly  $n$  distinct real solutions?

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