
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
Given that the parameter equations of curve $C$ are $\left\{\begin{array}{l}x=1+\sqrt{2}\cos\theta,\\ y=1+\sqrt{2}\sin\theta\end{array}\right.$ ($\theta$ is the parameter), the slope angle of line $l$ is $\alpha$, and it passes through point $P(0,1)$. <br/>$(1)$ Find the general equation of curve $C$ and the parameter equations of line $l$;<br/>$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and $\frac{1}{|PA|}+\frac{1}{|PB|}=\sqrt{5}$, find the slope angle $\alpha$ of line $l$.

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