
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 the quadratic function \( f(x) = ax^2 + bx + c \), where \( a \in \mathbf{N}^* \), \( c \geq 1 \), \( a + b + c \geq 1 \), and the equation \( ax^2 + bx + c = 0 \) has two distinct positive roots both less than 1, find the minimum value of \( a \).

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