
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
The quadratic function \( f(x) = ax^2 + bx + c \), where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \), satisfies the following conditions:
1. \( f(-1) = 0 \);
2. When \( x \in \mathbf{R} \), \( f(x) \geq x \);
3. When \( x \in (0, 2) \), \( f(x) \leq \frac{(x+1)^2}{4} \).

Given \( x_1, x_2, x_3 \in (0, 2) \) such that \( \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} = 3 \), find the minimum value of \( f(x_1) f(x_2) f(x_3) \).

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