
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 with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), its left focal point is $F_1(-1, 0)$, and vertex P on the ellipse satisfies $\angle PF_1O = 45^\circ$ (where O is the origin).
(1) Determine the values of $a$ and $b$;
(2) Given that line $l_1: y = kx + m_1$ intersects the ellipse at points A and B, and line $l_2: y = kx + m_2$ ($m_1 \neq m_2$) intersects the ellipse at points C and D, and $|AB| = |CD|$:
① Find the value of $m_1 + m_2$;
② Determine the maximum value of the area S of quadrilateral ABCD.

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