
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 numbers \( a, b, c \) are such that each of the two equations \( x^2 + bx + a = 0 \) and \( x^2 + cx + a = 1 \) has two integer roots, and all these roots are less than \( -1 \). Find the smallest value of \( a \).

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