
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
Archipelago consists of  $ n$  islands :  $ I_1,I_2,...,I_n$  and  $ a_1,a_2,...,a_n$  - number of the roads on each island.   $ a_1 \equal{} 55$ ,  $ a_k \equal{} a_{k \minus{} 1} \plus{} (k \minus{} 1)$ , ( $ k \equal{} 2,3,...,n$ )
a) Does there exist an island with 2008 roads?
b) Calculate  $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n.$ 

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