
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
For the quadratic function $y=ax^{2}-4ax-5\left(a\neq 0\right)$, the four conclusions are as follows: ① For any real number $m$, the function values corresponding to $x_{1}=2+m$ and $x_{2}=2-m$ are equal; ② Regardless of the value of $a$, the parabola must pass through two fixed points; ③ If the parabola intersects the $x$-axis at two distinct points $A$ and $B$, and $AB\leqslant 6$, then $a \lt -\frac{5}{4}$; ④ If for $3\leqslant x\leqslant 4$, there are $4$ integer values corresponding to $y$, then $-\frac{4}{3} \lt a\leqslant -1$ or $1\leqslant a \lt \frac{4}{3}$, the correct conclusion is ____ (fill in the number).

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