
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
The sequence \( \left(x_{n}\right)_{n=1}^{\infty} \) is defined recursively by
$$
x_{n+1} = \frac{x_{n} + (2 - \sqrt{3})}{1 - x_{n}(2 - \sqrt{3})}
$$
with \( x_{1} = 1 \). Determine the value of \( x_{1001} - x_{401} \).

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