
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 $f(x)=\frac{1}{2}[t\ln(x+2)-\ln(x-2)]$, and $f(x)\geqslant f(4)$ always holds.

(1) Find the value of $x$ when $f(x)$ takes the maximum value on $[3,7]$;

(2) Let $F(x)=a\ln(x-1)-f(x)$. If $f(x)$ is a monotonically increasing function, find the range of values for $a$.

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