
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 minimum value of the function $f(x)$ on the interval $[t,t+2](t > 0)$;
$(2)$ If there exists $x_{0}\in\left[ \frac{1}{e},e\right]$ (where $e$ is the base of the natural logarithm, $e=2.71828…$) such that the inequality $2f(x_{0})\geqslant g(x_{0})$ holds, find the range of the real number $a$.

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