
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 M: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{3}=1$$ ($a>0$) with one of its foci at F($-1,0$), and its left and right vertices are A and B, respectively. A line $l$ passing through point F intersects the ellipse M at points C and D.  
(1) When the slope angle of line $l$ is 45°, find the length of segment CD;  
(2) Let the areas of $\triangle ABD$ and $\triangle ABC$ be $S_1$ and $S_2$ respectively, find the maximum value of $|S_1-S_2|$.

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