
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 $C(t, \frac{3}{t})$ ($t\in\mathbb{R}, t\neq 0$) that passes through the origin $O$.  
(Ⅰ) Suppose the line $3x+y-4=0$ intersects the circle $C$ at points $M$ and $N$, and $|OM|=|ON|$, find the equation of circle $C$;  
(Ⅱ) Under the condition of (Ⅰ), let $B(0,2)$, and $P$, $Q$ be the moving points on the line $l: x+y+2=0$ and circle $C$ respectively, find the maximum value of $|PQ|-|PB|$ and the coordinates of point $P$ at this time.

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