
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
For each positive integer  $k$  greater than  $1$ , find the largest real number  $t$  such that the following hold:
Given  $n$  distinct points  $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$ ,  $\ldots$ ,  $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$  in  $\mathbb{R}^k$ , we define the score of the tuple  $a^{(i)}$  as
\[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\]
where  $\#S$  is the number of elements in set  $S$ , and  $\pi_j$  is the projection  $\mathbb{R}^k\to \mathbb{R}^{k-1}$  omitting the  $j$ -th coordinate. Then the  $t$ -th power mean of the scores of all  $a^{(i)}$ 's is at most  $n$ .

Note: The  $t$ -th power mean of positive real numbers  $x_1,\ldots,x_n$  is defined as 
\[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\]
when  $t\neq 0$ , and it is  $\sqrt[n]{x_1\cdots x_n}$  when  $t=0$ .

*Proposed by Cheng-Ying Chang and usjl*

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