
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 eccentricity $\frac{\sqrt{2}}{2}$, a circle with the diameter of the short axis of $C$ is tangent to the line $y=ax+6$.  
$(1)$ Find the equation of $C$;  
$(2)$ The line $l:y=k(x-1)(k\geqslant 0)$ intersects $C$ at points $A$ and $B$. Let $P$ be a point on $C$ such that a line parallel to the $x$-axis passing through $P$ intersects the line segment $AB$ at point $Q$. The slope of the line $OP$ (where $O$ is the origin) is denoted as $k'$. The area of $\triangle APQ$ is $S_{1}$ and the area of $\triangle BPQ$ is $S_{2}$. If $|AP|\cdot S_{2}=|BP|\cdot S_{1}$, determine whether $k\cdot k'$ is a constant, and explain the reason.

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