
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 the parametric equation of line $l$: 
$$\left\{ \begin{array}{l}
x = -1 + t \\
y = 2 + t
\end{array} \right.$$ 
($t$ being the parameter), and in the Cartesian coordinate system $xOy$ with origin $O$ as the pole and the positive x-axis as the polar axis, the polar equation of circle $C$ is $\rho^2 = 4 \sqrt{2} \rho \sin(\theta - \frac{\pi}{4}) - 6$.  
(Ⅰ) Find the Cartesian equations of line $l$ and circle $C$.  
(Ⅱ) If $A(-1, 2)$, and $P$, $Q$ are the two intersection points between line $l$ and circle $C$, calculate the length of $|PA| + |AQ|$.

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