
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
For the functions $f(x)= \frac{e^{x}-e^{-x}}{2}$, $g(x)= \frac{e^{x}+e^{-x}}{2}$, the following statements are given: $①$ The inequality $f(x) < g(x)$ holds true for all $x \in \mathbb{R}$; $②$ $[f(x)]^{2}+[g(x)]^{2} < 1$; $③f(2x)=2f(x)g(x)$. The correct statement(s) is/are __________.

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