
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]


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A rectangular prism is placed on a horizontal plane successively on three different faces. In these three cases, the ratios of the free surfaces of the prism are given by the numbers \( p, q, r \). Calculate the ratios of the edges of the rectangular prism. What restrictions must the numbers \( p, q, r \) satisfy for the problem to have a solution?

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