
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
A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth’s rotational pe­riod. It is at the precise distance of  $22,236$  miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to  $24$  hours. Be­ cause the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a tar­ get satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of  $22,236$  miles above the equator). Given that the radius of the Earth is  $3960$  miles, find the exact straight dis­tance between two neighboring stations. Write your answer in the form  $a + b\sqrt{c}$ , where  $a, b, c$  are integers and  $c > 0$  is square-free.

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