
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
The sequence of integers  $ a_1 $ ,  $ a_2 $ ,  $ \dots $  is defined as follows: $ a_1 = 1 $  and  $ n> 1 $ ,  $ a_ {n + 1} $  is the smallest integer greater than  $ a_n $  and such, that  $ a_i + a_j \neq 3a_k $  for any  $ i, j $  and  $ k $  from  $ \{1, 2, \dots, n + 1 \} $  are not necessarily different.
Define  $ a_ {2004} $ .

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