
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
Consider the quadratic polynomial \( P(x) = a x^2 + b x + c \) which has distinct positive roots. Vasya wrote four numbers on the board: the roots of \( P(x) \) and the roots of the polynomial \( Q(x) = c x^2 + b x + a \). What is the smallest integer value that the sum of these four numbers can have?

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