
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 $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=a\cos t \\ & y=2\sin t \end{cases}$ (where $t$ is the parameter, $a > 0$). With the origin as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of line $l$ is $\rho \cos \left( \theta +\dfrac{\pi }{4} \right)=-2\sqrt{2}$.

$(1)$ Let $P$ be a moving point on curve $C$. When $a=2\sqrt{3}$, find the maximum distance from point $P$ to line $l$;

$(2)$ If all points on curve $C$ are below and to the right of line $l$, find the range of values for $a$.

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