
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 center of circle $C$ is at $(2,0)$, and its radius is $\sqrt{2}$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. The parametric equation of line $l$ is: 
$$
\begin{cases}
x=-t \\
y=1+t
\end{cases} \quad (t \text{ is a parameter}).
$$

$(1)$ Find the polar coordinate equations of circle $C$ and line $l$;

$(2)$ The polar coordinates of point $P$ are $(1,\frac{\pi}{2})$, line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $|PA|+|PB|$.

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