
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, with the vertex at $(1,n)$ and $4a-2b+c=0$. The following four conclusions are given: ① If $n \gt 0$, then $abc \lt 0$; ② The equation $ax^{2}+bx+c=0$ must have a root at $x=4$; ③ If $n \lt 0$, then $a-3b+c \gt 0$; ④ Points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$ are two points on the parabola, with $x_{1} \lt x_{2}$, if $a(x_{1}+x_{2}-2) \lt 0$, then $y_{1} \gt y_{2}$; The correct sequence is ______.

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