
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 function $f(x) = x^2 - 1$ and the function $g(x) = a\ln x$ ($a \neq 0$).

(1) If the graphs of $f(x)$ and $g(x)$ have a common tangent line at the point $(1, 0)$, find the value of the real number $a$.

(2) Let $F(x) = f(x) - 2g(x)$, find the extreme values of the function $F(x)$.

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