
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 $f\left( x \right)=\frac{1-x}{ax}+\ln x$ where $a > 0$, and $e\approx 2.7$.

$(1)$ When $a=1$, find the equation of the tangent line to the function $f\left( x \right)$ at the point $\left( 1,f\left( 1 \right) \right)$ and the decreasing interval of $f\left( x \right)$.

$(2)$ If the function $f\left( x \right)$ is increasing on the interval $\left[ 2,+\infty \right)$, find the range of the real number $a$.

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