
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 line \( y = x - 1 \) intersects the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2} - 1} = 1 \) (where \( a > 1 \)) at points \( A \) and \( B \). If the circle with diameter \( AB \) passes through the left focus of the ellipse, find the value of \( a \).

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