
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
When the IMO is over and students want to relax, they all do the same thing:
    download movies from the internet. There is a positive number of rooms with internet
    routers at the hotel, and each student wants to download a positive number of bits. The
    load of a room is defined as the total number of bits to be downloaded from that room.
    Nobody likes slow internet, and in particular each student has a displeasure equal to the
    product of her number of bits and the load of her room. The misery of the group is
    defined as the sum of the students’ displeasures. 
    
    Right after the contest, students gather in the hotel lobby to decide who goes to which
    room. After much discussion they reach a balanced configuration: one for which no student
    can decrease her displeasure by unilaterally moving to another room. The misery
    of the group is computed to be  $M_1$ , and right when they seemed satisfied, Gugu arrived
    with a serendipitous smile and proposed another configuration that achieved misery  $M_2$ .
    What is the maximum value of  $M_1/M_2$  taken over all inputs to this problem?

*Proposed by Victor Reis (proglote), Brazil.*

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