
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
There are 5 families sharing a well, and each family has a rope of different length for drawing water, but all ropes are too short. The depth of the well is equal to twice the length of Family A's rope plus the length of Family B's rope, or three times the length of Family B's rope plus the length of Family C's rope, or four times the length of Family C's rope plus the length of Family D's rope, or five times the length of Family D's rope plus the length of Family E's rope, or six times the length of Family E's rope plus the length of Family A's rope. How deep is the well? What are the lengths of the ropes for each family?

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