
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
Let $f(x)=e^{x}-ax$ be a function defined on $R$. $(1)$ If there exists $x_{0}\in [1,+\infty)$ such that $f(x_{0}) < e-a$, find the range of real number $a$; $(2)$ Define: If real numbers $s$, $t$, $r$ satisfy $|s-r|\leq |t-r|$, then $s$ is said to be closer to $r$ than $t$. For $a$ in $(1)$ and $x\geq 1$, determine which is closer to $\ln x$ between $\frac{e}{x}$ and $e^{x-1}+a$, and explain the reason.

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