
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 $f(x) = e^x + a \cdot e^{-x}, x \in \mathbb{R}$.

(1) If $f(x)$ is monotonically increasing on $[0, +\infty)$, find the range of the real number $a$.

(2) If $a=1$, find the range of the real number $m$ such that $m[f(2x)+2] \geqslant f(x)+1$ holds true for all $x \in \mathbb{R}$.

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