
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 $f(x)=x\ln x$ and $g(x)=-x^{2}+ax-3$  
$(1)$ Find the equation of the tangent line to $f(x)$ at the point $(e,f(e))$  
$(2)$ If there exists $x\in[1,e]$ such that $2f(x)\geqslant g(x)$ always holds, find the range of values for $a$.

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