1999 Accord LX sedan-trouble starting

Hi, I've experienced this 4 or 5 times since I bought my Accord LX
new. About once a year it will not start. The starter cranks but the
engine does not start. I checked the spark plug wires to make sure
they are connected. The green security key light blinks on and off
while I'm trying to start the car. The gear shift cannot be moved--it
is stuck in park. In the past I just kept trying it and it finally
started, but this time it has not started and it's been 24 hours now.
The dealer told me the last time this happened that it might have
something to do with my key or the fact that I have a lot of keys on
my key chain that may interfere with the security system, but I tried
my wife's key and it doesn't help. Thanks for any help
Dennis