
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 base of a pyramid is a rhombus, one of whose angles is \(\alpha\). The lateral faces are equally inclined to the base. Through the midpoints of two adjacent sides of the base and the pyramid's apex, a plane is passed, which forms an angle \(\beta\) with the base. The area of the resultant cross-section is \(S\). Find the side length of the rhombus.

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