
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 functions $f(x) = \log_{\frac{1}{2}}(x^2 + 1)$ and $g(x) = x^2 - ax + 6$.
(I) If $g(x)$ is an even function, find the value of $a$ and write the increasing interval of $g(x)$;
(II) If the solution set of the inequality $g(x) < 0$ is $\{x \mid 2 < x < 3\}$, find the minimum value of $\frac{g(x)}{x - 1}$ when $x > 1$;
(III) For any $x_1 \in [1, +\infty)$ and $x_2 \in [-2, 4]$, if the inequality $f(x_1) \leq g(x_2)$ always holds, find the range of the real number $a$.

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