
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 curve $y=a^{x-1}+1(a>0$ and $a \neq 1)$ passes through the fixed point $\left(k,b\right)$, if $m+n=b-k$ and $m>0$, $n>0$, determine the minimum value of $\frac{9}{m}+\frac{1}{n}$.

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