
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 $f(x)=x\ln x+2+\frac{1}{a}$ and $g(x)=-x^{2}-bx-4$, where $x=\frac{5}{2}$ is a critical point of function $g(x)$. If for any $x_{1}\in [e^{-1}, 1]$, there exists a unique $x_{2}\in (-\infty, 3)$ such that $f(x_{1})=g(x_{2})$, determine the range of real number $a$.

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