
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
Given a sequence $\{a\_n\}$, define its "optimal value" as $H\_n = \frac{a\_1 + 2a\_2 + \dots + 2^{n-1}a\_n}{n}$. If the "optimal value" of a certain sequence $\{a\_n\}$ is $H\_n = 2^{n+1}$, let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n - kn\}$. If $S\_n \leq S\_5$ holds true for any $n$, find the range of values for the real number $k$.

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