
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 rectangular coordinate system, establish a polar coordinate system $(.$ with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. Given that point $A$ has polar coordinates $(\sqrt{2}, \frac{\pi}{4})$, and the polar equation of line $l$ is $\rho\cos(\theta - \frac{\pi}{4}) = \alpha$, with point $A$ lying on line $l$.

(1) Find the value of $\alpha$ and the rectangular equation of line $l$.
(2) The polar equation of circle $C$ is given by $\rho = 2\cos\alpha$. Determine the relationship between line $l$ and circle $C$.

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