
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 $A_n= \frac {a_{1}+2a_{2}+\ldots+2^{n-1}a_{n}}{n}$ as the "good number" of the sequence $\{a_n\}$. It is known that the "good number" $A_n=2^{n+1}$ for a certain sequence $\{a_n\}$. If the sum of the first $n$ terms of the sequence $\{a_n-kn\}$ is denoted by $S_n$, and $S_n\leqslant S_6$ holds for any $n\in\mathbb{N}^*$, determine the range of the real number $k$.

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