
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
The center of ellipse C is at the origin, and its foci, F<sub>1</sub> and F<sub>2</sub>, lie on the x-axis with a focal distance of 4. Let P be a moving point on ellipse C, and the interior angle ∠F<sub>1</sub>PF<sub>2</sub> of triangle PF<sub>1</sub>F<sub>2</sub> is maximized at $\frac{\pi}{2}$.
1. Determine the equation of ellipse C.
2. Is there a line intersecting ellipse C at points A and B, described by the equation $y=kx+m$ (where $k \in \mathbb{R}$), such that $|\vec{OA} + \vec{OB}| = |\vec{OA} - \vec{OB}|$? If such a line exists, find the range for the real number $m$; if not, provide an explanation.

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