
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 \( c \in \left(\frac{1}{2}, 1\right) \), find the smallest constant \( M \) such that for any integer \( n \geq 2 \) and real numbers \( 0 \leq a_1 \leq a_2 \leq \cdots \leq a_n \), if
$$
\frac{1}{n} \sum_{k=1}^{n} k a_{k} = c \sum_{k=1}^{n} a_{k},
$$
then it holds that
$$
\sum_{k=1}^{n} a_{k} \leq M \sum_{k=1}^{m} a_{k},
$$
where \( m = \lfloor c n \rfloor \) represents the greatest integer less than or equal to \( c n \).

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