
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 asymptote of the hyperbola $C: \frac{x^2}{a^2}- \frac{y^2}{b^2}=1$ ($a>0, b>0$) is perpendicular to the line $x+2y+1=0$. Let $F_1$ and $F_2$ be the foci of the hyperbola $C$ and let $A$ be a point on the hyperbola such that $|F_1A| = 2|F_2A|$. Determine the value of $\cos \angle AF_2F_1$.

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