
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with the right focus $F(1,0)$, and the lines connecting $F$ to the endpoints of the minor axis are perpendicular to each other.  
(Ⅰ) Find the standard equation of the ellipse $C$.  
(Ⅱ) Let point $Q$ be a point on the upper half of ellipse $C$, and let line passing through the origin $O$ and perpendicular to $QF$ intersect the line $y=2$ at point $P$. Find the minimum value of the area $S$ of triangle $OPQ$.

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