
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 which quadratic polynomials  $f(x)$  does there exist a quadratic polynomial  $g(x)$  such that the equations  $g(f(x)) = 0$  and  $f(x)g(x) = 0$  have the same roots, which are mutually distinct and form an arithmetic progression?

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