
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
If a line $l$ and a curve $C$ satisfy the following two conditions:

$(i)$ The line $l$ is tangent to the curve $C$ at point $P(x_{0},y_{0})$; $(ii)$ The curve $C$ is located on both sides of the line $l$ near point $P$, then the line $l$ is said to "cut through" the curve $C$ at point $P$.

The correct propositions are ______________ (Write down the numbers of all correct propositions).

$①$ The line $l$: $y=0$ "cuts through" the curve $C$: $y=x^{3}$ at point $P(0,0)$;  
$②$ The line $l$: $x=-1$ "cuts through" the curve $C$: $y=(x+1)^{3}$ at point $P(-1,0)$;  
$③$ The line $l$: $y=x$ "cuts through" the curve $C$: $y=\sin x$ at point $P(0,0)$;  
$④$ The line $l$: $y=x$ "cuts through" the curve $C$: $y=\tan x$ at point $P(0,0)$;  
$⑤$ The line $l$: $y=x-1$ "cuts through" the curve $C$: $y=\ln x$ at point $P(1,0)$.

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