
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
<u>Set 1</u>**1.1** How many positive integer divisors are there of  $2^2 \cdot 3^3 \cdot 5^4$ ?**1.2** Let  $T$  be the answer from the previous problem. For how many integers  $n$  between  $1$  and  $T$  (inclusive) is  $\frac{(n)(n - 1)(n - 2)}{12}$  an integer?**1.3** Let  $T$  be the answer from the previous problem. Find  $\frac{lcm(T, 36)}{gcd(T, 36)}$ .


PS. You should use hide for answers. 

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