
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 that the function $f(x)$ is a differentiable function defined on $\mathbb{R}$, for any real number $x$, we have $\frac{f(-x)}{f(x)}=e^{2x}$. When $x < 0$, $f(x)+f'(x) > 0$. Determine the range of real number $a$ such that $e^{a}f(2a+1) \geq f(a+1)$.

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