
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 $f(x) = ax^2 + bx + c$ ($ac \neq 0$), and $g(x) = cx^2 + bx + a$  
① If $f(x)$ has no real roots, then $g(x) > 0$ for all $x \in \mathbb{R}$.  
② If $f(x)$ has exactly one real root, then $g(x)$ must have two real roots.  
③ If the equation $f(x) = 0$ has two distinct real roots, then the equation $g(x) = 0$ cannot have no solution.  
The number of true statements is __.

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