
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
For the numerical sequence \( x_{0}, x_{1}, \ldots, x_{n}, x_{n+1}, \ldots \), the relations \( 2x_{n} = x_{0} + x_{1} + \cdots + x_{n-1} - x_{n} \) hold true for all \( n = 0, 1, 2, \ldots \). Find each term \( x_{n} \) of this sequence and the values of the sums \( S_{n} = x_{0} + x_{1} + \cdots + x_{n} \).

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