
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 ellipse $x^{2}+ \frac{y^{2}}{4}=1$, $A$ and $B$ are the left and right vertices of the ellipse, and $P$ is a point on the ellipse that does not coincide with $A$ or $B$. The angles of inclination of $PA$ and $PB$ are $α$ and $β$ respectively. Find the value of $\frac{\cos (α-β)}{\cos (α+β)}$.

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