
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  $f(x)$  be a quadratic function with integer coefficients. If we know that  $f(0)$ ,  $f(3)$  and  $f(4)$  are all different and elements of the set  $\{2, 20, 202, 2022\}$ , determine all possible values of  $f(1)$ .

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