
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 that the right focus of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) is $F$, and the line passing through point $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of $AB$ is $(1, -\frac{1}{2})$, and the angle of inclination of line $AB$ is $45^\circ$, determine the equation of the ellipse.

A) $\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1$
B) $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
C) $\frac{2x^{2}}{9} + \frac{4y^{2}}{9} = 1$
D) $\frac{x^{2}}{9} + \frac{2y^{2}}{9} = 1$

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