
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 plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=1+\frac{1}{2}t,}\\{y=-1+\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the circle $C$ is $\rho ^{2}-2\rho \cos \theta -4\rho \sin \theta +1=0$. <br/>$(1)$ Find the general equation of circle $C$ and the rectangular coordinate equation of line $l$; <br/>$(2)$ Given the point $P(1,-1)$, 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}
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