Re-enlistment

I signed up for a year last August. I had planned to sign up for three but the recruiter said if I sign up for one, I can re-enlist for three then get a bonus. I could not get a bonus for enlisting because I had prior service and needed job training. So I wrote a note to the retention specialist (the person in charge of re-enlistment) in our brigade.
Here is her answer to my email:
Your entire email sounds like a test question! I love it...I will do some
research for you and then I will get back to you with an answer.
Have a great day!

And here's the test question:
I was at Aberdeen for training during the last drill, but wanted to contact
you before the next one April 18-20.
I am now an MOSQ 63J10. [Meaning I completed the training.]
My enlistment ends August 15, 2008.
I will be 55 years old on May 2.
At the time of my enlistment on August 16, 2007, I had 11 years, 2 months,
2 days of service including 6 years and 7 months active, three years active
reserve and the rest inactive.
SO
Can I re-enlist for six years just after June 15 getting me over 18 years at
the end of that enlistment at age 61 allowing me to retire because I will
be both over 60 and over 18 years?
Will I have to enlist in an active duty unit to get the retirement?
Let me know.

It does sort of sound like a word problem on a test.
I'll let you know what she says.