
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{2}}{2}$. The chord intercepted by the ellipse from the line $l$, which passes through point $F_2$ and is perpendicular to the x-axis, has a length of $\sqrt{2}$.

(Ⅰ) Find the equation of ellipse $C$.

(Ⅱ) Let $A$ and $B$ be two points on $C$, and the perpendicular bisector of segment $AB$ intersects $C$ at points $P$ and $Q$. If the midpoint $M$ of segment $AB$ lies on line $l$, find the range of values for $\overrightarrow{F_1P} \cdot \overrightarrow{F_1Q}$.

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