
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 $f(x)= \frac{1}{2}e^{2x}+a$, there exists a point $P$ on the graph of $f(x)$, and there exists a point $Q$ on the graph of $g(x)=\ln x$ such that the points $P$ and $Q$ are symmetric with respect to the line $y=x$. Find the range of the real number $a$ satisfying the above requirement.

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