
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 $E$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with foci $F_1$ and $F_2$, and eccentricity $\dfrac{\sqrt{2}}{2}$, the line $l$: $x + 2y - 2 = 0$ intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively.  
(Ⅰ) If point $A$ is a vertex of the ellipse $E$, find the equation of the ellipse $E$;  
(Ⅱ) If there exists a point $P$ on segment $AB$ such that $|PF_1 + PF_2| = 2a$, find the range of values for $a$.

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