
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 foci $F_{1}$ and $F_{2}$, and eccentricity $\frac{\sqrt{2}}{2}$. Let $P$ be a point on $C$, and the area of $\triangle PF_{1}F_{2}$ is maximized at $4$. <br/>$(Ⅰ)$ Find the equation of $C$;<br/>$(Ⅱ)$ If the line $y=kx+t\left(t\neq 0\right)$ intersects $C$ at points $A$ and $B$, and a line passing through point $B$ perpendicular to $y=3$ intersects at point $D$, and the intersection point of line $AD$ and the $y$-axis is a fixed point $Q$, find the value of $t$ and the coordinates of the fixed point $Q$.

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