
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
The following are four propositions:
1. A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(2)<f(3)$, then the function $f(x)$ is not monotonically decreasing on $\mathbb{R}$.
2. Let $A=\{1,4\}$, $B=\{1,-1,2,-2\}$, and $f: x \rightarrow \sqrt[7]{x^2}$. Then $f$ is a mapping from $A$ to $B$.
3. The graph of the function $f(x)=2^{-x}$ is shifted two units to the right and one unit downward. The resulting graph corresponds to the function $g(x)=2^{-(x-2)}-1$.
4. The equation $|2^x-1|=a$ (where $a$ is a constant) must have two different real solutions when $a>0$.
The correct proposition number(s) are: (Answer with the number(s))

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