
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 $F(a,b)= \begin{cases} a & \text{if } a\leqslant b, \\ b & \text{if } a > b. \end{cases}$ Given that the domain of functions $f(x)$ and $g(x)$ is $\mathbb{R}$, which of the following propositions are true? ______ (Write down all the numbers of the true propositions)
1. If both $f(x)$ and $g(x)$ are odd functions, then the function $F(f(x),g(x))$ is an odd function;
2. If both $f(x)$ and $g(x)$ are even functions, then the function $F(f(x),g(x))$ is an even function;
3. If both $f(x)$ and $g(x)$ are increasing functions, then the function $F(f(x),g(x))$ is an increasing function;
4. If both $f(x)$ and $g(x)$ are decreasing functions, then the function $F(f(x),g(x))$ is a decreasing function.

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