
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
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b \cdot \cos C + c \cdot \cos B = 4a \cdot \sin^2 \frac{A}{2}$.

(Ⅰ) Find angle $A$;

(Ⅱ) If $a=3$, $b=\sqrt{6}$, and the perpendicular bisector of $BC$ intersects $AB$ at point $D$ and $BC$ at point $E$, find the area of quadrilateral $ACED$.

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