
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 $f(x) = x^2 - ax - a\ln(x-1)$ where $a \in \mathbb{R}$,
1. Find the extreme values of the function $f(x)$ when $a = 1$.
2. Determine the monotonic intervals of the function $f(x)$.
3. Explain whether there exists a real number $a$ ($a \geq 1$) such that the graph of $y = f(x)$ has no common points with $y = \frac{5}{8} + \ln 2$.

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