
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

Given a quadratic polynomial \( f(x) = ax^2 + bx + c \) such that

\[ f\left( \frac{a - b - c}{2a} \right) = f\left( \frac{c - a - b}{2a} \right) = 0 \]

find the value of the product \( f(-1) \cdot f(1) \).

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