
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
In the quadratic function $y=ax^{2}+bx+c$ (where $a$, $b$, and $c$ are constants and $a\neq 0$), the corresponding values of $x$ and $y$ are shown in the table below:<br/>

| $x$ | $-1$ | $0$ | $1$ | $3$ |
|-----|-----|----|----|----|
| $y$ | $-1$ | $3$ | $5$ | $3$ |

The following conclusions are given:<br/>
① $ac \lt 0$;<br/>
② When $x \gt 1$, the value of $y$ decreases as $x$ increases;<br/>
③ $3$ is a root of the equation $ax^{2}+\left(b-1\right)x+c=0$;<br/>
④ When $-1 \lt x \lt 3$, $ax^{2}+bx+c \gt 0$.<br/>

Which of the following conclusions are correct?

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