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Speaking ahead of the second Test against New Zealand starting at Trent Bridge on Friday, 29-year-old left-hander Lees reflected on his score of 20 - including four fours - in the fourth innings of the series opener at Lord's, which England won by five wickets.
"Last week was probably the most fluent innings I've had to date," Lees said. "I was pleased in the manner I played but the obvious thing is that I have to take that and turn it into a substantial innings.
"To be praised for a 20 is probably bittersweet. If you can get a good 20, you know you can probably make 60, 70, 80. I keep getting in and out, which is frustrating.
"When you open the batting you face the best bowlers, sometimes at the worst times. To get through a spell or get through a tough day of cricket is something that I've always quite enjoyed… it's that old Yorkshire stubbornness.
"There's always going to be speculation about the top order in England… I just never really wanted to shy away from the challenge of it."
Since making his debut against West Indies in the Caribbean in March after being called up in the aftermath of England's dismal Ashes defeat, Lees has made 171 Test runs at a strike rate of 30.05 with a highest score of 31.
He was bowled shouldering arms to a Kyle Jamieson nip-backer which gave New Zealand their initial breakthrough defending a lead of 276 before Joe Root's unbeaten 115 led England to victory early on Sunday.
"It's not putting pressure on myself to make a double-hundred this game, but I'm aware I'd love to make a good score," Lees said. "It's trusting how I play and I played nicely in that second innings, so if I can take that forward and play in that manner over a period of time, the law of averages suggest I should get that contribution. That's what I'm striving towards.
"The obvious thing is if you've been selected as a batter, I think you always need to show some worth otherwise you know, you're not going to get re-selected. I think the bigger picture for me is that I want to keep playing in a way in which I like to play, which is a way in which Brendon [McCullum, the England Test head coach] and Ben want us to play as a team and buy into that team ethos and identity.
"And if I believe that, if we do that, and if I do that, I think there's no reason over the period of the next couple of games why I can't get a score."
Stokes bowled sparingly during England's training session in Nottingham on Wednesday after appearing to be in some discomfort with his side, although the problem did not curb his batting in the nets.
Spinner Jack Leach, meanwhile, bowled in the middle for the first time since he was concussed while fielding on the first day of the opening Test, forming part of a phased return to action that means he remains in doubt for the second match. Matt Parkinson, who was Leach's concussion substitute at Lord's, has held his place in the squad, with Craig Overton offering another pace option.