
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
Let the function $f(x)=e^{2x}+ae^{x}$, where $a\in \mathbb{R}$.

$(1)$ When $a=-4$; find the intervals of monotonicity for $f(x)$;

$(2)$ If for all $x\in \mathbb{R}$, $f(x)\geqslant a^{2}x$ always holds, find the range of the real number $a$.

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