
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
In the Cartesian coordinate system $xOy$, the coordinates of the three vertices of $\triangle ABC$ are $A(4,2)$, $B(3,0)$, and $C(4,0)$. The graph of the inverse proportion function $y=\frac{k}{x}$ passes through the midpoint $D$ of $AC$.  
$(1)$ Let the intersection point of the line $AB$ and the $y$-axis be point $P$. Find the length of $OP$.  
$(2)$ Given that point $F$ is symmetric to point $B$ with respect to point $E(2,2)$. Determine whether point $F$ lies on the graph of the inverse proportion function and provide a reason.

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