
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
For the quadratic equation $ax^{2}+bx+c=0\left(a\neq 0\right)$, the following propositions are given: If ①$a+b+c=0$, then $b^{2}-4ac\geqslant 0$; ②If the roots of the equation $ax^{2}+bx+c=0$ are $-1$ and $2$, then $2a+c=0$; ③If the equation $ax^{2}+c=0$ has two distinct real roots, then the equation $ax^{2}+bx+c=0$ must have two distinct real roots; ④If the equation $ax^{2}+bx+c=0$ has two equal real roots, then $ax^{2}+bx+c=1$ has no real roots. The true propositions are ______.

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