
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases}x=1- \frac{1}{2}t, \\ y= \frac{ \sqrt{3}}{2}t,\end{cases}$ ($t$ is the parameter). With the origin as the pole and the positive $x$-axis as the polar axis, the equation of circle $C$ is $\rho=2 \sqrt{3} \sin \theta$.

$(1)$ Write the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$;

$(2)$ If the Cartesian coordinates of point $P$ are $(1,0)$, and circle $C$ intersects line $l$ at points $A$ and $B$, find the value of $|PA|+|PB|$.

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