Looking for visa info on visiting Ecuador as a family

We are a family with 3 children hoping to spend 4-6 months this winter in Ecuador.  I'm trying to make sense of the laws on tourist visas versus just getting an extension.  My understanding at this point is that we have to pay $450 per person ($2250 total) for a tourist visa to be in the country six months.  Or we can go for 90 days for free and then request an extension once we are there for 90 more days.

My question is:  What is the process for requesting an extension?  Where do we go to do that?  What information do we need?  And how much is the cost for doing that?

I've been online for about 3 hours so far trying to gather all of this info.  I've called the Consulate in Los Angeles but received a message machine.  I've chatted online and corresponded by email.  But I'm still having a difficult time getting clear answers.  I'm hoping someone here can answer my questions.

Heather