
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
Define $H_n = \frac{a_1 + 2a_2 + \ldots + 2^{n-1}a_n}{n}$ as the average value of the sequence $\{a_n\}$. Given that the average value of the sequence $\{b_n\}$, denoted as $H_n$, equals $2^{n+1}$, and let the partial sum of the first $n$ terms of the sequence $\{b_n - kn\}$ be $S_n$. If $S_n \leq S_3$ holds for any positive integer $n$, then the range of the real number $k$ is ______.

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