
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
Let \( m \) be a natural number greater than 1. The sequence \( x_{0}, x_{1}, x_{2}, \ldots \) is defined by

\[ x_{i} = \begin{cases}2^{i}, & \text{for } 0 \leq i \leq m-1 \\ \sum_{j=1}^{m} x_{i-j}, & \text{for } i \geq m\end{cases} \]

Find the largest \( k \) such that there are \( k \) consecutive sequential elements that are all divisible by \( m \).

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