
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 an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with left and right vertices $A_{1}$, $A_{2}$, and left and right foci $F_{1}$, $F_{2}$, a line $l$ passing through point $F_{1}$ with a slope of $\frac{\sqrt{5}}{2}$ intersects the ellipse $C$ at points $D$ and $E$. If the perimeter of $\triangle DEF_{2}$ is $8$, and the distance from point $F_{2}$ to line $l$ is $\frac{2\sqrt{5}}{3}$.<br/>$(1)$ Find the equation of the ellipse $C$.<br/>$(2)$ Given point $M$ is any point on the ellipse $C$, lines $MA_{1}$, $MA_{2}$ intersect the line $x=4$ at points $P$ and $Q$ respectively. Determine if there exists a fixed point $T$ outside the ellipse $C$ such that $\angle PTQ$ is always a right angle. If it exists, find the coordinates of the fixed point $T$; if not, explain the reason.

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