
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 $f(x)=\frac{x}{e^{2x}}$, where $e=2.71828...$ is the base of the natural logarithm.
(I) Find the monotonic intervals and the maximum value of $f(x)$.
(II) Let $g(x)=\frac{x}{e^{2x}}+m$. If the tangent line to the curve $g(x)$ at the point $(-\frac{1}{2},g(-\frac{1}{2}))$ passes through the point $(1,3e)$, find the value of $m$.

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