
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
Assume the function $f(x) = x|x| + bx + c$ ($b, c \in \mathbb{R}$). Consider the following four propositions:
1. If $c = 0$, then $f(x)$ is an odd function.
2. If $b = 0$, then the function $f(x)$ is increasing over $\mathbb{R}$.
3. The graph of $y = f(x)$ is centrally symmetric about the point $(0, c)$.
4. The equation $f(x) = 0$ has at most two real roots.
Determine the correct propositions.

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