
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 that the extreme value point $x_{0} \in (\frac{1}{2}, 1)$ of the function $f(x) = e^x + e^{-x} - a \ln x$ ($a \in \mathbb{N}, a \geq 2$) exists.
(I) Find the value of $a$;
(II) If the inequality $f(x) \geq b$ ($b \in \mathbb{Z}$) always holds, find the maximum value of $b$.

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