
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
Given that the function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are constants) is an odd function, and it satisfies $f(1) = b + c$, $f(2) = 4a + 2b + c$.  
(1) Find the values of $a$, $b$, and $c$;  
(2) Determine the monotonicity of the function $f(x)$ on the interval $(0, +\infty)$ and explain the reason;  
(3) Find the minimum value of the function $f(x)$ on the interval $(0, +\infty)$.

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