
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 $f(x) = a^2e^{-2x} + a(2x+1)e^{-x} + x^2 + x$ ($a > 0$), if its minimum value is $\ln^2{a} + 3\ln{a} + 2$, then the range of $a$ is $\_\_\_\_\_\_$.

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