
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
For every positive integer \( n \), define \( a_{n} \) as the last digit of the sum of the digits of the number formed by writing "2005" \( n \) times consecutively. For example, \(\mathrm{a}_{1}=7\) and \(\mathrm{a}_{2}=4\).

a) What are the positive integers \( n \) such that \( a_{n}=0 \)?

b) Calculate \( a_{1}+a_{2}+\cdots+a_{2005} \).

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