
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  $M=\{1,2,\dots,49\}$  be the set of the first  $49$  positive integers. Determine the maximum integer  $k$  such that the set  $M$  has a subset of  $k$  elements such that there is no  $6$  consecutive integers in such subset. For this value of  $k$ , find the number of subsets of  $M$  with  $k$  elements with the given property.

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