
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 that the hyperbola C:\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0) has an asymptote with equation $y=\frac{\sqrt{5}}{2}x$, and shares a common focus with the ellipse $\frac{{{x}^{2}}}{12}+\frac{{{y}^{2}}}{3}=1$, determine the equation of C.

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