
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, the parametric equation of line $l$ is $\begin{cases} x=t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$, where $t$ is the parameter, $\alpha\in(0, \frac{\pi}{2})$. Furthermore, by taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho\sin^2\theta+2\sin\theta=\rho$, where $\rho\geqslant 0$, $\theta\in\mathbb{R}$. Line $l$ intersects curve $C$ at points $P$ and $Q$.
$(1)$ Find the value of $\overrightarrow{OP}\cdot\overrightarrow{OQ}$;
$(2)$ Given point $A(0,1)$, and $|AP|=2|AQ|$, find the general equation of line $l$.

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