
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  $p = 491$  be prime. Let  $S$  be the set of ordered  $k$ -tuples of nonnegative integers that are less than  $p$ . We say that a function  $f\colon S \to S$  is \emph{ $k$ -murine} if, for all  $u,v\in S$ ,  $\langle f(u), f(v)\rangle \equiv \langle u,v\rangle \pmod p$ , where  $\langle(a_1,\dots ,a_k) , (b_1, \dots , b_k)\rangle = a_1b_1+ \dots +a_kb_k$  for any  $(a_1, \dots a_k), (b_1, \dots b_k) \in S$ .
	Let  $m(k)$  be the number of  $k$ -murine functions. Compute the remainder when  $m(1) + m(2) + m(3) + \cdots + m(p)$  is divided by  $488$ .

*Proposed by Brandon Wang*

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