
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 $\left\{a_{n}\right\}$ where $a_{1}=1$, $a_{2}=2$, and $a_{n}a_{n+1}a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ for all $n$, and $a_{n+1}a_{n+2} \neq 1$, find $S_{1999} = \sum_{n=1}^{1999} a_{n} = \quad$.

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