
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 $f(x)=\log_{a}x+m$ ($a > 0$ and $a \neq 1$) whose graph passes through the point $(8,2)$, and the point $P(3,-1)$ is symmetric to the point $Q$ with respect to the line $x=2$ on the graph of $f(x)$.

$(1)$ Find the expression of the function $f(x)$;

$(2)$ Let $g(x)=2f(x)-f(x-1)$, find the minimum value of $g(x)$ and the value of $x$ when $g(x)$ reaches its minimum.

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