
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
Let  $n$  and  $k$  be two integers which are greater than  $1$ . Let  $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$  be non-negative real numbers such that
i)  $a_1\ge a_2\ge\ldots\ge a_n$  and  $a_1+a_2+\ldots+a_n=1$ ;
ii) For any integer  $m\in\{1,2,\ldots,n\}$ , we have that  $c_1+c_2+\ldots+c_m\le m^k$ .
Find the maximum of  $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$ .

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