
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 that the vertex coordinates of the parabola $y=ax^{2}+bx+c$ (where $a$, $b$, and $c$ are constants) are $(-\frac{1}{2},m)$, where $m \gt 0$, and one intersection point with the $x$-axis lies between $(0,0)$ and $(1,0)$, then the following conclusions are made:<br/>①$b \lt 0$;<br/>②$2b+c \gt 0$;<br/>③If the parabola passes through the points $(-2,y_{1})$ and $(2,y_{2})$, then $y_{1} \gt y_{2}$;<br/>④If the quadratic equation in $x$, $ax^{2}+bx+c-2=0$, has no real roots, then $m \gt 2$.<br/>The correct conclusions are _____. (Fill in the correct numbers)

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