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Mamardashvili played the first half for the Reds in their opening pre-season 3-1 friendly win at Championship side Preston North End on Sunday afternoon.
The 24-year-old Georgia international arrived at the club at the start of the month after Liverpool had agreed a £29million deal with Valencia last August.
Alisson has been the undisputed first-choice between the sticks for Liverpool since signing from Roma seven years ago, although he has been sidelined by various injury problems for spells in each of the last two seasons.
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And while acknowledging the Brazilian's continued impact, Mamardashvili wants to write his own name in the lengthy annals of Anfield goalkeeping history.
"What I can tell about Ali, he's one of the best goalkeepers in the world right now and I am so happy to work with him," he said.
It's a huge moment for me because joining a club like Liverpool, it's not only a transfer, it's a dream come true for me.
But now I truly understand what it means to be Liverpool's player.
"When I was a child I always watched some videos on YouTube of goalkeepers, and I remember some Liverpool goalkeepers as well.
Liverpool has a big goalkeeper legacy and I want to try to write my own chapter in this goalkeepers' legacy."
Among those Mamardashvili worked with at Valencia was former Liverpool goalkeeping coach Jose Manuel Ochotorena, who joined the Reds along with Rafa Benitez in 2004 before leaving three years later when replaced by Xavi Valero, who this summer has returned to his old role at Anfield.
And Mamardashvili has revealed the message he was given by Ochotorena when interest from Liverpool became apparent.
"Ochotorena is an amazing person and an amazing coach," said the Georgian, speaking to LFC TV.
"Actually, he was one of the first who I talked (to) about Liverpool's offer and he directly told me 'just sign the contract' because he knows how big (a) club Liverpool is, and I just want to tell him thank you.
"I don't like too much talking about myself, but I think I'm a very calm, concentrated goalkeeper and I'm very fast and quick in the goal.
"I know the responsibilities that come with wearing this (Liverpool) shirt.
I'm here just to give my best in every training session, every match to help the team and earn the trust of our fans."
Sunday's game at Preston was Liverpool's first since the death of Diogo Jota and his brother Andre Silva following a car crash in Spain earlier this month.
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And Mamardashvili said: "When I first arrived here it was a really tough moment for the club.
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