Pellet stove venting question

Hi newbie hear with some questions. I recently picked up a used breckwell P4000 Vermont from a friend that never installed it. I want to replace my wood stove in my basement that vents into a 6" thimble in the wall that goes through the concrete foundation & into a cement block clay lined chimney. I know I should run 4" pipe or line up to the top of the chimney which is about 18' but here is my question. The stove came with duravent pro piping to vent through an exterior wall. I would like to use this piping to connect my stove into the existing flue & then tie into the liner or pipe I buy to go up the chimney. Is it ok to have this 3" pipe one of the stove and up to the flue & then use 4" going up the chimney? I would like to avoid buying it all again. The run would be into a T then vertical about 3" the an elbow and about 24" into the existing thimble and chimney where I would connect an increaser tee and then 4" liner or pipe going up 18' to chimney cap.

Thanks in advance for any help!

According to the manual after 15' EVL it recommends 4" pipe. I would definitely be over this amount I'm just trying to avoid having to buy pipe I already have if it's not going to make a huge difference. The EVL from stove to liner would be like 18 including the T connecting to the liner going up the masonry chimney. So to me it sounds close to what the owners manual recommends & probably save me buying $200-300 in pipe I already have in 3" size. Thanks for the info guys! Great site here.

Thanks for the info guys. I think I'm going to go this route I'm just deciding if I can use a liner all the way to the wall thimble and then just use one of those liner to duravent pro wall/thimble adapters or if I need to install a T in the chimney. I'm not sure how flexible those SS liners are. It would need to basically make a 90 degree turn at the bottom of the chimney to go through the 6" thimble. Anyone know if that might be possible? Thanks

Thanks! I think I will go the route with the T then liner up to a section of rigid pipe at the termination. Then I will need to see if I need a fresh air hookup. I'm in the basement so I would have to run out the side then up an through the sill. I'm going to try without it first. Thanks again for the advice!