
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 $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ ($a>b>0$) with foci $F_1(-c,0)$ and $F_2(c,0)$ ($c>0$), a line passing through the point $E\left( \frac {a^{2}}{c},0\right)$ intersects the ellipse at points A and B, with $F_1A\parallel F_2B$ and $|F_1A|=2|F_2B|$.  
(1) Find the eccentricity of the ellipse.  
(2) Find the slope of the line AB.  
(3) Assume point C is symmetric to point A about the origin and there is a point H(m,n) ($m \neq 0$) on line $F_2B$ and on the circumcircle of $\triangle AF_1C$. Find the value of $\frac {n}{m}$.

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