
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 parabola $y=ax^{2}+bx+c$ (where $a$, $b$, and $c$ are constants), and its graph passes through point $A(2,0)$, with the origin as $O$.<br/>① If $b=-2a$, then the parabola must pass through the origin;<br/>② If $c\neq 4a$, then the parabola must have two distinct points in common with the $x$-axis;<br/>③ If the parabola intersects the $x$-axis at point $B$ (not coinciding with $A$), intersects the $y$-axis at point $C$, and $OB=OC$, then $a=-\frac{1}{2}$;<br/>④ Points $M(x_{1}, y_{1})$ and $N(x_{2}, y_{2})$ are on the parabola. If $x_{1} \gt x_{2} \gt -1$, and $y_{1} \gt y_{2}$ whenever $x_{1} \gt x_{2} \gt -1$, then $8a+c\leqslant 0$. The correct conclusions are ______ (fill in the numbers).

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