
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 Cartesian coordinate system, if both $x$ and $y$ are integers, then the point $(x,y)$ is called an integer point. Among the following statements, the correct ones are __________ (write down the numbers of all correct statements).

$①$ There exists such a line that is neither parallel to the coordinate axes nor passes through any integer points.  
$②$ If both $k$ and $b$ are irrational numbers, then the line $y=kx+b$ does not pass through any integer points.  
$③$ A line $l$ passes through infinitely many integer points if and only if $l$ passes through two different integer points.  
$④$ The necessary and sufficient condition for the line $y=kx+b$ to pass through infinitely many integer points is that both $k$ and $b$ are rational numbers.  
$⑤$ There exists a line that passes through exactly one integer point.

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