
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
7. Peggy picks three positive integers between  $1$  and  $25$ , inclusive, and tells us the following information about those numbers:


- Exactly one of them is a multiple of  $2$ ;
- Exactly one of them is a multiple of  $3$ ;
- Exactly one of them is a multiple of  $5$ ;
- Exactly one of them is a multiple of  $7$ ;
- Exactly one of them is a multiple of  $11$ .

What is the maximum possible sum of the integers that Peggy picked?

8. What is the largest positive integer  $k$  such that  $2^k$  divides  $2^{4^8}+8^{2^4}+4^{8^2}$ ?

9. Find the smallest integer  $n$  such that  $n$  is the sum of  $7$  consecutive positive integers and the sum of  $12$  consecutive positive integers.

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