
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
For two quadratic trinomials  $P(x)$  and  $Q(x)$  there is a linear function  $\ell(x)$  such that  $P(x)=Q(\ell(x))$  for all real  $x$ . How many such linear functions  $\ell(x)$  can exist?

*(A. Golovanov)*

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