
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 the quadratic function $f(x) = ax^2 + bx + c$, consider the following statements:
1. If $f(p) = q$, $f(q) = p$ ($p \neq q$), then $f(p + q) = -(p + q)$;
2. If $f(p) = f(q)$ ($p \neq q$), then $f(p + q) = c$;
3. If $f(p + q) = c$ ($p \neq q$), then $p + q = 0$ or $f(p) = f(q)$.

The statements that are definitely correct are __________ (write down the numbers of all correct statements).

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