
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 $xoy$, the parameteric equation of line $l$ is given by $\begin{cases}x=1+2t \\ y=2+t\end{cases}\left(t \text{ is the parameter}\right)$. In the polar coordinate system with origin $O$ and positive $x$-axis as the polar axis, the polar equation of circle $C$ is given by $\rho =2$.

(1) Write the general equation of line $l$ and the standard equation of circle $C$.
(2) Given point $P(-1,1)$, line $l$ intersects circle $C$ at points $A$ and $B$. Find the value of $\left|\left|PA\right|-\left|PB\right|\right|$.

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