
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 $y=\log_{a}(x-2)+1$ $(a > 0$ and $a\neq 1)$, its graph always passes through point $A$. If point $A$ is on the line $mx+ny-1=0$, where $m,n\in \mathbb{R}_{+}$, find the minimum value of $\frac{3}{m}+\frac{1}{n}$.

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