
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 $f(x)=\log_{a}x+b$ ($a > 0$, $a \neq 1$) whose graph passes through the points $(8,2)$ and $(1,-1)$  
$(1)$ Find the expression for $f(x)$  
$(2)$ If $[f(x)]^{2}=3f(x)$, find the value of the real number $x$  
$(3)$ Let $y=g(x)=2f(x+1)-f(x)$, find the minimum value of $y=g(x)$, and the value of $x$ when $y$ is at its minimum.

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