
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 a rectangular coordinate system with its origin $O$ as the pole, the positive semi-axis of the $x$-axis as the polar axis, and both coordinate systems having equal unit lengths, the parameteric equations of line $l$ are $\begin{cases} x = 1 + \sqrt{3}t \\ y = 1 + t \end{cases}$, where $t$ is the parameter, and the polar equation of circle $C$ is $ρ = 2$.

1. Write the Cartesian equations for line $l$ and circle $C$.
2. Let $P(1, 1)$. Find the value of $|PA| - |PB|$, where $A$ and $B$ are the intersection points of line $l$ and circle $C$.

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