
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 Cartesian coordinate system, with the origin as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar coordinates of point $A$ are $\left( \sqrt{2}, \frac{\pi}{4} \right)$, and the polar equation of line $l$ is $\rho\cos \left( \theta- \frac{\pi}{4} \right)=a$, and point $A$ lies on line $l$.

$(1)$ Find the value of $a$ and the Cartesian equation of line $l$;

$(2)$ The parametric equation of circle $C$ is $\begin{cases} x=1+\cos \alpha, \\ y=\sin \alpha \end{cases}$ (where $\alpha$ is the parameter), determine the positional relationship between line $l$ and circle $C$.

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