
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
Let $a_{1}, a_{2}, \cdots, a_{100}$ be an arbitrary permutation of $\{1901, 1902, \cdots, 2000\}$. Consider the partial sum sequence $S_{1}=a_{1}$, $S_{2}=a_{1}+a_{2}$, $S_{3}=a_{1}+a_{2}+a_{3}, \cdots, S_{100}=a_{1}+a_{2}+\cdots+a_{100}$. If no term in the sequence $S_{j} (1 \leqslant j \leqslant 100)$ is divisible by 3, how many such sequences exist?
(Canadian Olympiad 2000)

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