
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 circle with center at point $A(m, \frac{2}{m})$ ($m \in \mathbb{R}$ and $m > 0$) intersects the $x$-axis at points $O$ and $B$, and intersects the $y$-axis at points $O$ and $C$, where $O$ is the origin.  
$(1)$ When $m=2$, find the standard equation of circle $A$;  
$(2)$ As $m$ varies, is the area of $\triangle OBC$ a constant? If yes, find the constant; if no, explain why;  
$(3)$ Suppose line $l: 2x+y-4=0$ intersects circle $A$ at points $P$ and $Q$, and $|OP|=|OQ|$, find the value of $|PQ|$.

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