
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 fixed integer greater than  $1$ . The sequence  $x_0$ ,  $x_1$ ,  $x_2$ ,  $\ldots$  is defined as follows: 
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest  $k$  for which the sequence contains  $k$  consecutive terms divisible by  $m$  .

*Proposed by Marcin Kuczma, Poland*

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