
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, suppose a curve $C_1$ is given by $x^2+y^2=1$. All points on curve $C_1$ have their $x$ and $y$ coordinates stretched by a factor of $\sqrt{2}$ and $\sqrt{3}$, respectively, resulting in a new curve $C_2$.
$(1)$ Write down the parametric equations for curve $C_2$.
$(2)$ Find the maximum distance from a point on curve $C_2$ to the line $l$: $x+y-4\sqrt{5}=0$.

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