
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 circle $M$: $(x+ \sqrt{3}a)^{2}+y^{2}=16a^{2}(a > 0)$ and a fixed point $N(\sqrt{3}a,0)$, point $P$ is a moving point on circle $M$, and point $G$ is on line $MP$ satisfying $|GP|=|GN|$. The trajectory of point $G$ is curve $C$.

$(1)$ Find the equation of curve $C$;

$(2)$ If the symmetric point of $A(1,0)$ with respect to the line $x+y-t=0(t > 0)$ is on curve $C$, find the range of values for $a$.

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