
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
For the sequence $\{a_n\}$, define $H_n= \frac {a_1+2a_2+\ldots+2^{n-1}a_n}{n}$ as the "excellent value" of $\{a_n\}$. Now, it is known that the "excellent value" $H_n=2^{n+1}$ for some sequence $\{a_n\}$. Let the sum of the first $n$ terms of the sequence $\{a_n-kn\}$ be $S_n$. If $S_n\leqslant S_6$ holds for any $n$, then the range of the real number $k$ is \_\_\_\_\_\_.

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