
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
Define the function $f(x)$ on $[0,+\infty)$ that satisfies $2f(x)+f''(x)= \frac{\sqrt{x}}{e^x}$, where $f''(x)$ is the derivative of $f(x)$, and $f\left( \frac{1}{2}\right)= \frac{1}{2\sqrt{2e}}$. For any positive numbers $a$ and $b$, determine the range of $\theta$ if $f(\sin \theta)\leq \frac{1}{a^2} + \frac{1}{e^2b^2} + \frac{ab}{64}$.

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