Find all triplets (X,Y,Z) of positive integers that satisfy this system of equations:
lcm(X,Y,Z) + Y +Z = 426, and:
X + gcd(X,Y,Z) + Z = 66, and:
We can lift you up, and help your day.
Sarcastically taken we can go either way.
Short as a whole, lengthy in the middle.
Six letter word brings laughter to this riddle.
What are we?