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Panevėžys is a town in Lithuania.
To catch a ride to Latvia, you need to take city bus Nr15 towards Piniava and get out at the last stop (Called "Piniava"). Walk towards the river, cross the bridge and find a place to stand safely and hitch-hike. If you are only passing Panevėžys by, you may be dropped out at the circuit; it is a bit more difficult to hitchhike there.
Take buses Nr.1, 3, 13, 14 towards Kniaudiškių st and get out at the last stop close to a big supermarket "Babilonas". Then cross the street called Klaipėdos. You can try to hitch-hike there and start walking a bit more out of the city.
Take bus Nr.16 and go till the last bus stop "Prekybos bazė", cross the street and walk out the city. It's quite difficult to hitch-hike there because it's still city street. However, you can easily get a lift till the junction with the bypass where the highway starts and it is a great place for catching a ride to Vilnius, Kaunas and other towns too.
Also you can walk till the very end of Panevėžys where the roads starts to look new. Follow the road south until there is a sign marking the end of the city. Right after there is a bus stop with plenty of space to stop. Most cars are already going a bit faster, but at least you avoid allmost all of the local village-bound traffic.
For n0id this worked quite fast after standing for a long time closer to the city where most people driving by just indicated that they were turning from this road and not going far.