
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 $l$ is $\begin{cases}x=4t \\ y=4t+a\end{cases}$ (where $t$ is the parameter and $a\in \mathbb{R}$), and the polar equation of circle $C$ is $\rho=4\cos \theta-4\sin \theta$.  

(1) Convert the parametric equation of line $l$ into a standard equation, and convert the polar equation of circle $C$ into a Cartesian coordinate equation;  

(2) If there are exactly three points on circle $C$ 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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