
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 $g(x)=a-{{x}^{2}}$ $(\frac{1}{e}\leqslant x\leqslant e$, where $e$ is the base of the natural logarithm) and $h(x)=2\ln x$, there exist points on the graph of $g(x)$ and $h(x)$ that are symmetric about the $x$-axis. The range of the real number $a$ is ______.

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