
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
Suppose that  $c\in\left(\frac{1}{2},1\right)$ . Find the least  $M$  such that for every integer  $n\ge 2$  and real numbers  $0<a_1\le a_2\le\ldots \le a_n$ , if  $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$ , then we always have that  $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$  where  $m=[cn]$ 

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