
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 $f(x)$ reaches 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 $a$.

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