
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
In the sequence \(\{a_n\}\), \(a_1 = -1\), \(a_2 = 1\), and \(a_3 = -2\), for all \(n \in \mathbb{N_+}\) it holds that \(a_n a_{n+1} a_{n+2} a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3}\) and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\). Determine the value of the sum of the first 4321 terms of the sequence \(S_{4321}\).

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