
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) opening downwards, passing through points $A(-1,0)$ and $B(m,0)$, where $1 < m < 2$. The following four conclusions are given:<br/>① $b \gt 0$;<br/>② If $m=\frac{3}{2}$, then $3a+2c \lt 0$;<br/>③ If points $M(x_{1}, y_{1})$ and $N(x_{2}, y_{2})$ are on the parabola, with $x_{1} \lt x_{2}$ and $x_{1}+x_{2} \gt 1$, then $y_{1} \gt y_{2}$;<br/>④ When $a\leqslant -1$, the quadratic equation in $x$, $ax^{2}+bx+c=1$, must have two distinct real roots.<br/>Which of the following is correct?____(Fill in the numbers).

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