
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 fixed point \( A(1,1) \) and the focus \( F \) of the ellipse \( \frac{x^2}{8} + \frac{y^2}{4} = 1 \), where point \( P \) lies on the ellipse. Find the maximum and minimum values of \( |PF| + |PA| \), and the coordinates of point \( P \) when these extreme values are attained.

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