
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 polynomial with integer coefficients such that  $f(15) f(21) f(35) - 10$  is divisible by  $105$ . Given  $f(-34) = 2014$  and  $f(0) \ge 0$ , find the smallest possible value of  $f(0)$ .

*Proposed by Michael Kural and Evan Chen*

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