
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 $$\frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1\ (0＜b＜2)$$, with its foci at F<sub>1</sub> and F<sub>2</sub>, a line l passing through F<sub>1</sub> intersects the ellipse at points A and B. If the maximum value of |BF<sub>2</sub>|+|AF<sub>2</sub>| is 6, find the value of $b$.

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