
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 the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) has an eccentricity of $\frac{\sqrt{2}}{2}$. Let point $F$ be the left focus of the ellipse, point $A$ be the right vertex of the ellipse, and point $B$ be the top vertex of the ellipse, with the area of triangle $\triangle ABF$ being $\frac{\sqrt{2}+1}{2}$.
(Ⅰ) Find the equation of the ellipse $E$.
(Ⅱ) If the line $l: x - 2y - 1 = 0$ intersects the ellipse $E$ at points $P$ and $Q$, find the perimeter and the area of triangle $\triangle FPQ$.

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