
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
On Binary Island, residents communicate using special paper. Each piece of paper is a  $1 \times n$  row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has  $10$  random consecutive squares each of which is randomly colored red or green.

Malmer and Weven would like to develop a scheme that allows them to send messages of length  $2016$  between one another. They would like to be able to send any message of length  $2016$ , and they want their scheme to work with perfect accuracy. What is the smallest value of  $n$  for which they can develop such a strategy?

*Note that when sending a message, one can see which 10 squares are colored and what colors they are. One also knows on which square the message begins, and on which square the message ends.*


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