
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 $f(x)=\begin{cases} & {(\dfrac{1}{2})}^{x},x\geqslant 0 \\ & {(\dfrac{1}{e})}^{x},x < 0 \end{cases}$, determine the range of the real number $a$ such that the inequality $f(2x+a)\geqslant {[f(x)]}^{3}$ holds for any $x\in [1-2a,1+2a]$.

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