
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 equation of one asymptote of the hyperbola $\frac {y^{2}}{a^{2}} - \frac {x^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) is $y= \frac { \sqrt {3}}{3}x$, and the length of the chord passing through the focus and perpendicular to the y-axis is 6.

(1) Find the equation of the hyperbola;  
(2) A line with an inclination of 45° passing through the lower focus of the hyperbola intersects the curve at points M and N. Find the length of MN.

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