
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
k is a fixed real number such that \( \frac{1}{2} < k < 1 \). What is the smallest real h such that if positive reals \( x_1 \leq x_2 \leq \ldots \leq x_n \) satisfy \(\frac{x_1 + 2x_2 + 3x_3 + \ldots + nx_n }{n} = k (x_1 + x_2 + \ldots + x_n) \), then they also satisfy \( x_1 + x_2 + \ldots + x_n \leq h (x_1 + x_2 + \ldots + x_m) \), where \( m = \lfloor kn \rfloor \).

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