
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 $y=\log _{a}(x-3)+1\left(a \gt 0,a\neq 1\right)$, its graph always passes through a fixed point $A$. If point $A$ lies on the graph of the linear function $y=\frac{1}{m}x+\frac{1}{n}$, where $m \gt 0$, $n \gt 0$, then the minimum value of $m+n$ is ____.

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