
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
Determine the following for the line $l$ and the circle $C$:
Line $l$: $\begin{cases} x=a+4t \\ y=-1-2t \end{cases}$ (where $t$ is a parameter),
Circle $C$: $\rho=2\sqrt{2}\cos\left(\theta+ \frac{\pi}{4}\right)$ (the polar axis coincides with the non-negative half-axis of the x-axis, and the unit length is the same).
(1) Find the distance from the center of circle $C$ to line $l$.
(2) If the chord cut by line $l$ on circle $C$ has a length of $\frac{6\sqrt{5}}{5}$, find the value of the real number $a$.

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