
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 $f\left(x\right)=\log _{a}(x-1)+2(a \gt 0$, and $a\neq 1)$, and its graph passes through a fixed point $P$.  
$(1)$ Find the coordinates of $P$;  
$(2)$ If the graph of $f\left(x\right)$ on $\left[2,4\right]$ always lies below the line $y=-x+8$, find the range of values for $a$.

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