
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  $p$  be a prime and let  $f(x) = ax^2 + bx + c$  be a quadratic polynomial with integer coefficients such that  $0 < a, b, c \le p$ . Suppose  $f(x)$  is divisible by  $p$  whenever  $x$  is a positive integer. Find all possible values of  $a + b + c$ .

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