
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
Is there a function $f(x)$ that simultaneously satisfies the following three conditions: (1) $f(x) > 0$ for all $x \in \mathbb{N}$; (2) $f(a+b) = f(a)f(b)$ for all $a, b \in \mathbb{N}$; and (3) $f(2) = 4$? If such a function exists, find an explicit formula for $f(x)$. If not, explain why.

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