
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 C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}$$\=1 (a>b>0) with left and right foci $F\_1$ and $F\_2$, respectively. The ratio of the major axis length to the focal distance of ellipse C is $$\sqrt {2}$$:1. The line l passing through $F\_2$(3,0) intersects C at points A and B.
(1) When the slope of l is 1, find the area of triangle $F\_1AB$.
(2) When the perpendicular bisector of segment AB has the smallest intercept on the y-axis, find the equation of line l.

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