
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 a square $ABCD$ with side length $1$, divide side $AB$ into $n$ equal parts, where $n\geqslant 2$. Point $M$ is the point on $AB$ closest to point $A$. After cutting off the square with side length $AM$ from square $ABCD$, the area of the remaining part is denoted as $S$. Let $S=S_{1}\cdot S_{2}\cdot \ldots \cdot S_{n}$.  
$(1)$ When $n=2023$, find the value of $S$;  
$(2)$ If the graph of the function $y=kx$ (where $k\neq 0$) intersects the inverse proportional function graph at the point $(n,S-\frac{1}{2})$ at points $A$ and $B$. A line parallel to the $x$-axis passing through point $A$ and a line parallel to the $y$-axis passing through point $B$ intersect at point $P$, determine whether the area of $\triangle ABP$ is a constant value. If it is, find the value; if not, explain why.

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