
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 equation of ellipse C<sub>1</sub> is $\frac {x^{2}}{4} + y^{2} = 1$, and the left and right foci of hyperbola C<sub>2</sub> are respectively the left and right vertices of ellipse C<sub>1</sub>, while the left and right vertices of hyperbola C<sub>2</sub> are respectively the left and right foci of C<sub>1</sub>.
(1) Find the equation of hyperbola C<sub>2</sub>;
(2) If the line $l: y = kx + \sqrt{2}$ always intersects hyperbola C<sub>2</sub> at two distinct points A and B, and $\vec{OA} \cdot \vec{OB} > 2$ (where O is the origin), find the range of values for $k$.

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