
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 the curve $C: y=4-x^{2}$ intersects the $x$-axis at two distinct points $A$ and $B$ (point $A$ is to the left of point $B$), and a point $P(t,0)$ lies on the line segment $AB$ (excluding the endpoints). A perpendicular line is drawn from point $P$ to the $x$-axis, intersecting the curve $C$ at point $Q$. <br/>$(Ⅰ)$ If triangle $APQ$ is an isosceles right triangle, find the area of triangle $APQ$; <br/>$(Ⅱ)$ Let the area of triangle $APQ$ be denoted as $S(t)$, find the maximum value of $S(t)$.

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