
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 $f(x)=2\ln x$ and $g(x)=a-x^{2}$ ($-e \leqslant x \leqslant - \frac {1}{e}$), where $e$ is the base of the natural logarithm, find the range of the real number $a$ such that it is always possible to find a point $P$ on the graph of $f(x)$ and a point $Q$ on the graph of $g(x)$ that are symmetric about the origin.

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