
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 rectangular coordinate system $xOy$, the parametric equation of line $l$ is $$\begin{cases} x=t\cos\alpha \\ y=1+t\sin\alpha \end{cases}$$ ($t$ is the parameter, $\alpha\in[0,π)$). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho^2=2\rho\cos\theta+3$.

1. Find the general equation of line $l$ and the rectangular coordinate equation of curve $C$.
2. If line $l$ intersects curve $C$ at points $A$ and $B$, and $|AB|=2\sqrt{2}$, find the equation of line $l$.

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