
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ passing through points $A(2,0)$ and $B(0,1)$.

(Ⅰ) Find the equation of ellipse $C$ and its eccentricity;

(Ⅱ) Suppose point $Q$ is on ellipse $C$. Is there a point $P$ on the line $x+y-4=0$ such that the quadrilateral $PAQB$ is a parallelogram? If yes, find the coordinates of point $P$; if no, explain why.

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