
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 (C) with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, one of its foci coincides with the focus of the parabola $y^2 = 4x$, denoted as $F\_1$. The other focus of the ellipse is $F\_2$. Let $M$ be any point on the ellipse. If the maximum area of the triangle $\triangle MF\_1F\_2$ is $1$:

1. Find the equation of the ellipse (C).
2. Let line $l: y = kx + m$ (which does not pass through the origin) intersect the ellipse (C) at two distinct points $A$ and $B$. If the slope of line $l$ is the geometric mean of the slopes of lines $OA$ and $OB$, find the range of values for the area of $\triangle OAB$.

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