
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 rich emir was admiring a new jewel, a small golden plate in the shape of an equilateral triangle, decorated with diamonds. He noticed that the shadow of the plate forms a right triangle, with the hypotenuse being the true length of each side of the plate. 

What is the angle between the plane of the plate and the flat surface of the sand? Calculate the cosine of this angle.

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