
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
Consider the sequence \(\{a_n\}\):
\[1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, \cdots,\]

This sequence is known as the "fractal sequence" and is generated as follows:
First, we have \(a_1 = 1\). Next, copy this term 1 and add its successor 2, getting \(a_2 = 1, a_3 = 2\);

Then, copy all the previous terms \(\{1, 1, 2\}\) and add the successor of 2, which is 3, resulting in \(a_4 = 1, a_5 = 1, a_6 = 2, a_7 = 3\);

Next, copy all the previous terms \(\{1, 1, 2, 1, 1, 2, 3\}\) and add the successor of 3, which is 4, resulting in the first 15 terms:
\[1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4 \text{;}\]

Continue this pattern.
Find \(a_{2000}\) and the sum of the first 2000 terms \(S_{2000}\).

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