
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 a hyperbola with its focus on the $y$-axis, its asymptotes are $y=± \frac { \sqrt {3}}{3}x$, and $c=2$. It is known that point $A(1, \frac {1}{2})$ is on the hyperbola.

(I) Find the standard equation of the hyperbola;

(II) Line $L$ passes through point $A$ and intersects the hyperbola at points $M$ and $N$. Point $A$ is the midpoint of segment $MN$. Find the equation of line $L$.

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