
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
A positive integer  $n$  is said to be  $m$ -free if  $n \leq m!$  and  $\gcd(i,n)=1$  for each  $i=1,2,...,m$ . Let  $\mathcal{S}_k$  denote the sum of the squares of all the  $k$ -free integers. Find the remainder when  $\mathcal{S}_7-\mathcal{S}_6$  is divided by  $1000$ .

*Proposed by **FedeX333X***

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