
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 $t$;  
$(2)$ Find the value of $x$ for which $f(x)$ attains its maximum value on the interval $[3,7]$;  
$(3)$ Let $F(x)=a\ln (x-1)-f(x)$. If $F(x)$ is a monotonically increasing function, find the range of values for $a$.

Please choose one of the questions 22 or 23 to answer. If more than one is answered, only the first one will be scored. Make sure to indicate the question number when answering.

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