
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
For a positive integer \( a \) and integers \( b \) and \( c \), the quadratic equation \( ax^2 + bx + c = 0 \) has two roots \( \alpha \) and \( \beta \) that satisfy \( 0 < \alpha < \beta < 1 \). Find the minimum value of \( a \).

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