
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
Consider two functions defined on \( \mathbb{R} \): \( f(x) = \frac{\pi^x - \pi^{-x}}{2} \) and \( g(x) = \frac{\pi^x + \pi^{-x}}{2} \) (where \( \pi \) is the constant pi, \( \pi = 3.1415926\ldots \)). The following statements are proposed:
① \( f(x) \) is an odd function, and \( g(x) \) is an even function;
② \( f(x) \) is an increasing function on \( \mathbb{R} \), and \( g(x) \) is a decreasing function on \( \mathbb{R} \);
③ \( f(x) \) has no maximum or minimum value, while \( g(x) \) has a minimum value but no maximum value;
④ For any \( x \in \mathbb{R} \), it holds that \( f(2x) = 2f(x)g(x) \);
⑤ \( f(x) \) has zeros, while \( g(x) \) has no zeros.
Identify all correct statements.

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