
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 functions $f\left(x\right)=x^{2}+ax+1$ and $g\left(x\right)=\ln x-a\left(a\in R\right)$.  
$(1)$ When $a=1$, find the extreme value of the function $h\left(x\right)=f\left(x\right)-g\left(x\right)$.  
$(2)$ If there exists a line that is tangent to the graphs of both functions $f\left(x\right)$ and $g\left(x\right)$, find the range of real numbers for $a$.

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