
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 $y=ax^{2}-(5a+1)x+4a+4$, the following statements are made:<br/>① If $a \lt -1$, then the graph of the quadratic function intersects the negative half-axis of the $y$-axis:<br/>② If $a \gt 0$, when $1\leqslant x\leqslant 2$, $y$ has a maximum value of $3$;<br/>③ If $a$ is an integer, and the graph of the quadratic function has two common points with the $x$-axis that are both integer points, then the value of $a$ can only be $1$;<br/>④ If $a \lt 0$, and $A(2$,$y_{1})$, $B(3$,$y_{2})$, $C(4$,$y_{3})$ are three points on the graph of the function, then $y_{1} \gt y_{2} \gt y_{3}$.<br/>The correct statements are ______. (Only fill in the numbers)

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