
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 $y=\log_{a}(x-1)+2$ $(a > 0, a \neq 1)$ always passes through a fixed point $A$. If the line $mx+ny=2$, where $m$ and $n$ are positive real numbers, passes through point $A$, determine the minimum value of $\frac{1}{m} + \frac{2}{n}$.

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