
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  $f(x) = (x - 5)(x - 12)$  and  $g(x) = (x - 6)(x - 10)$ . 
Find the sum of all integers  $n$  such that  $\frac{f(g(n))}{f(n)^2}$  is defined and an integer.


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