
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
Anne and Bill decide to play a game together. At the beginning, they chose a positive integer  $n$ ; then, starting from a positive integer  $\mathcal{N}_0$ , Anne subtracts to  $\mathcal{N}_0$  an integer  $k$ -th power (possibly  $0$ ) of  $n$  less than or equal to  $\mathcal{N}_0$ . The resulting number  $\mathcal{N}_1=\mathcal{N}_0-n^k$  is then passed to Bill, who repeats the same process starting from  $\mathcal{N}_1$ : he subtracts to  $\mathcal{N}_1$  an integer  $j$ -th power of  $n$  less than or equal to  $\mathcal{N}_1$ , and he then gives the resulting number  $\mathcal{N}_2=\mathcal{N}_1-n^j$  to Anne. The game continues like that until one player gets  $0$  as the result of his operation, winning the game. For each  $1\leq n \leq 1000$ , let  $f(n)$  be the number of integers  $1\leq \mathcal{N}_0\leq 5000$  such that Anne has a winning strategy starting from them. For how many values of  $n$  we have that  $f(n)\geq 2520$ ?

*Proposed by **FedeX333X***

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