
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
In an apartment, the length of room I is equal to the length of room II, the width of room II is equal to the width of room III, and the diagonal of room I is equal to the diagonal of room III. (The rooms' bases are naturally rectangular.) The areas of the rooms are $a$, $b$, and $c$ square meters, respectively. Calculate the dimensions of each room.

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