
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 $f\left(x\right)=a\left(x+\ln x\right)-xe^{x}$.
1. When $a=0$, find the equation of the tangent line to the function $f\left(x\right)$ at $x=1$.
2. If $f\left(x\right) \lt 0$ holds for $x\in \left[1,+\infty \right)$, determine the range of real numbers for $a$.
3. When $a=1$, find the maximum value of the function $f\left(x\right)$.

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