
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 a quadratic equation in $x$, $ax^{2}+bx+c=0$ where $a\neq 0$, the following statements are made:<br/>① If $a-b+c=0$, then $b^{2}-4ac\geqslant 0$;<br/>② If the roots of the equation $ax^{2}+bx+c=0$ are $1$ and $2$, then $2a-c=0$;<br/>③ If the equation $ax^{2}+c=0$ has two distinct real roots, then the equation $ax^{2}+bx+c=0$ must have real roots;<br/>④ If $b=2a+c$, then the equation has two distinct real roots.<br/>Which of the following statements are correct? ______. (Fill in the corresponding numbers)

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