
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 $ax^{2}+bx+c=0$ where $a\neq 0$.<br/>① If the two roots of the equation are $1$ and $2$, then $2a-c=0$;<br/>② If $b=2a+c$, then the quadratic equation $ax^{2}+bx+c=0$ has two distinct real roots;<br/>③ If $m$ is a root of the equation $ax^{2}+bx+c=0$, then $b^{2}-4ac=\left(2am+b\right)^{2}$ must hold true.<br/>Determine whether the above statements are correct and explain the reasons.

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