
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
The parametric equation of line $l$ is $\begin{cases} x=2t \\ y=4t+a \end{cases}$ (where $t$ is the parameter), and the polar equation of circle $C$ is $\rho=4\cos\theta-4\sin\theta$.  
(1) Convert the polar equation of circle $C$ into a Cartesian coordinate equation;  
(2) If there are exactly three points on the circle that are at a distance of $\sqrt{2}$ from line $l$, find the value of the real number $a$.

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