
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 a theater, there are $k$ rows of seats. 770 spectators came to the theater and sat in their seats (some seats may have been left unoccupied). After the intermission, all spectators forgot their original seats and sat down differently. What is the largest value of $k$ for which it is guaranteed that there are 4 spectators who sat in the same row both before and after the intermission?

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