
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
Among the following four propositions:

1. Given the function $f(x+1)=x^{2}$, then $f(e)=(e-1)^{2}$;
2. The range of the function $f(x)$ is $(-2,2)$, then the range of the function $f(x+2)$ is $(-4,0)$;
3. The graph of the function $y=2x (x \in \mathbb{N})$ is a straight line;
4. Given that $f(x)$ and $g(x)$ are two functions defined on $\mathbb{R}$, for any $x$, $y \in \mathbb{R}$ they satisfy the relation $f(x+y)+f(x-y)=2f(x) \cdot g(y)$, and $f(0)=0$, but $f(x) \cdot g(x) \neq 0$ when $x \neq 0$, then functions $f(x)$ and $g(x)$ are both odd functions.

Determine the incorrect proposition(s).

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