
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
Given the sequence \(\left\{a_n\right\}\):
\[1,1,2,1,1,2,3,1,1,2,1,1,2,3,4, \cdots,\]
called the "fractal sequence," construct it as follows:

1. Start with \(a_1 = 1\).
2. Copy the initial term and append its successor 2, resulting in \(a_2 = 1, a_3 = 2\).
3. Copy all previous terms \(1,1,2\) and append the successor of 2, which is 3, resulting in \(a_4 = 1, a_5 = 1, a_6 = 2, a_7 = 3\).
4. Continue this process by copying all previous terms \(1,1,2,1,1,2,3\) and appending the successor of 3, which is 4, giving the first 15 terms:
\[1,1,2,1,1,2,3,1,1,2,1,1,2,3,4\]

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

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