
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 point P (-2 $$\sqrt{2}$$, 0) as the left vertex of an ellipse $$\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}} = 1$$ (with a > b > 0), and a tangent line to the circle $$x^{2}+y^{2}=4$$ passing through P with tangency points A and B. If the line AB passes exactly through the left focus F of the ellipse C, derive the value of $$a^{2}+b^{2}$$.

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