
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
You have the first $n$ positive integers written on a sheet of paper as $1, 2, \cdots, n$. In the first step, you cross out the first four numbers, $1, 2, 3, 4$, and write their sum, 10, at the end of the sequence. In the second step, you cross out the next four numbers, $5, 6, 7, 8$, and write their sum, 26, at the end of the sequence. This process continues with each step involving crossing out the next four numbers and writing their sum at the end.

(1) If there is only one number left at the end, what necessary and sufficient condition must $n$ satisfy?

(2) Given $n = 2002$, until only one number is left, what is the total sum of all the numbers written (including the original sequence $1, 2, \cdots, 2002$)?

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