
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 functions $f(x)=\ln x- \frac{1}{x}$ and $g(x)=ax+b$.
(1) Find the range of values for the real number $a$ such that the function $h(x)=f(x)-g(x)$ is monotonically increasing on $(0,+\infty)$.
(2) If the line $g(x)=ax+b$ is a tangent line to the graph of the function $f(x)=\ln x- \frac{1}{x}$, find the minimum value of $a+b$.

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