
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 $$C: \frac{x^2}{a^2} + \frac{y^2}{3} = 1 (a > \sqrt{3})$$ has an eccentricity of $$\frac{\sqrt{2}}{2}$$, and its right vertex is A.  
(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) If a line $l$ passes through the left focus $F_1$ of C and intersects C at points B and D, find the maximum area of $\triangle ABD$ and the corresponding equation of line $l$.

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