
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
Suppose \( x_0, x_1, x_2, \cdots \) is a sequence of numbers such that \( x_0 = 1000 \), and
\[ x_n = -\frac{1000}{n}\left(x_0 + x_1 + x_2 + \cdots + x_{n-1}\right) \]
for all \( n \geq 1 \). Find the value of
\[ \frac{1}{2^{2}} x_0 + \frac{1}{2} x_1 + x_2 + 2 x_3 + 2^2 x_4 + \cdots + 2^{997} x_{999} + 2^{998} x_{1000} \]

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