
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
A group of  $n > 1$  pirates of different age owned total of  $2009$  coins. Initially each pirate (except the youngest one) had one coin more than the next younger.
a) Find all possible values of  $n$ .
b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates. If  $n = 7$ , find the largest possible number of coins a pirate can have after several days.

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