
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 $M=\left\{1,2, \cdots, 2^{m} n\right\}$ (where $m$ and $n$ are positive integers) be a set consisting of continuous $2^{m}n$ positive integers. Find the minimum value of the positive integer $k$ such that any $k$-element subset of $M$ contains $m+1$ numbers $a_1, a_2, \cdots, a_{m+1}$ that satisfy $a_i$ divides $a_{i+1}$ for $i=1,2, \cdots, m$.

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