
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
Define the function $f(x)$ on $\mathbb{R}$ that satisfies both of the following conditions: (i) For any real numbers $a$ and $b$ in the domain, $f(a+b) = f(a)f(b)$; (ii) When $x > 0$, $f(x) > 1$. Then,
(1) Provide a specific example of a function that meets the above conditions;
(2) Find the value of $f(0)$;
(3) Compare $f(1)$ and $f(3)$ and explain the reasoning.

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