
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 ellipse $C: \frac{x^2}{4} + \frac{y^2}{3} = 1$ and the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a,b > 0)$ have the same foci, and the eccentricities of the two curves are reciprocal to each other. Find the sine of the angle of inclination of the asymptotes of the hyperbola.

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