
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
Selected Exercise $4-4$: Polar Coordinates and Parametric Equations

In a polar coordinate system, the polar equation of the curve $C$ is $\rho = 4 \cos \theta$, and the polar equation of the line $l$ is $\sqrt{2} \rho \sin \left( \theta + \frac{\pi}{4} \right) = 1$. Point $Q$ with polar coordinates $\left(\rho, \frac{\pi}{2}\right)$ lies on line $l$. Establish a rectangular coordinate system $xOy$ with the pole as the coordinate origin $O$ and the polar axis as the positive $x$-axis, and use the same unit length for both coordinate systems.

(I) Find the rectangular equations for the curve $C$ and the line $l$;

(II) If line $l$ intersects curve $C$ at two distinct points $A$ and $B$, find the value of $|QA| + |QB|$.

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