
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
The parabola $y=ax^{2}+bx+c(a$,$b$,$c$ are constants, $c \lt 0)$ passes through the three points $\left(1,1\right)$, $\left(m,0\right)$, $\left(n,0\right)$, where $n\geqslant 3$. The following four conclusions are given:<br/>①$b \lt 0$;<br/>②$4ac-b^{2} \lt 4a$;<br/>③When $n=3$, if the point $\left(2,t\right)$ lies on the parabola, then $t \gt 1$;<br/>④If the quadratic equation in $x$, $ax^{2}+bx+c=x$, has two equal real roots, then $0＜m≤\frac{1}{3}$. The correct conclusions are______$(fill in the numbers)$.

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