• England, 1-0 up in the series, will start as marginal favourites to earn their seventh successive Test win.
• England were left with 11 overs to bat at the of the day
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England need 210 more runs to win the second Test after finally dismissing New Zealand for 483 on day four in Wellington.
Kane Williamson made a faultless 132 and Tom Blundell 90 as the Black Caps ground their way back into the game after being asked to follow-on.
It left England needing to pull off the highest run-chase by any team that has enforced the follow-on in a Test, while only three other teams have ever made more than New Zealand following-on against England.
The toil of England's bowlers on a docile pitch was made harder by captain Ben Stokes not joining the attack because of his troublesome left knee.
Even Harry Brook was called on to bowl his medium pace, remarkably claiming Williamson as his first Test wicket and sparking a New Zealand collapse of 5-28.
England were left with 11 overs to bat at the of the day, in which time the under-pressure Zak Crawley was bowled through the gate by Tim Southee for a skittish 24.
Ben Duckett remains on 23 in the company of nightwatchman Ollie Robinson on 1, with England 48-1.
Bizarre day sets up superb finale
From looking one-sided when England asked New Zealand to bat again on Saturday, this Test has the potential to provide a superb finale on day five.
Day four veered from the sublime of Williamson's hundred to the ridiculous of Brook bowling, a move that ultimately limited England's target.
It would not have been England's plan to face such a large chase, but the pitch remains good for batting and they have recent pedigree for successful pursuits.
With England repeatedly stating their commitment to not draw matches, it leaves the prospect of a historic final day whatever the outcome. The tourists will either set a new record chase or become the first England team to lose a Test after enforcing the follow-on.
England, 1-0 up in the series, will start as marginal favourites to earn their seventh successive Test win.
New Zealand, though, have given themselves a chance of protecting an unbeaten record in home series that stretches back six years.
Wonderful Williamson back to his best
Williamson had managed only 10 previous runs in the series, but after becoming New Zealand's all-time leading run-scorer in the first over of the day, went on to compile the most serene century.
Always playing late and with the softest of hands, the former captain clipped off his pads and repeatedly punched off the back foot.
He added 75 for the fifth wicket with Daryl Mitchell, who made a run-a-ball 54, then piled on a stand of 158 with Blundell.
While Williamson went to a chanceless century, his 26th in Tests, by cutting Broad for four, Blundell survived sharp chances to James Anderson at backward square leg on 30 and bowler Leach on 70.
Williamson was devastated to fall to Brook's bowling after a tickle down the leg side was detected on review.
The New Zealand collapse left Blundell with only last man Neil Wagner for company.
Aiming for his second century of the series, he fell 10 runs short with a charge and swipe at Leach that ended in the hands of slip.
Foakes and Leach keep England going
This was a third long day in the field for England's weary bowlers, who cycled through any number of tactics and plans.
Their toil would have been longer had it not been for wicketkeeper Ben Foakes, who again proved his worth as one of the premier glovemen on the planet.
He came up to the stumps for the pace of Stuart Broad, a move that instantly drew a miscue from Mitchell into the hands of Joe Root running back from first slip.
Foakes' movement to take the catch of Williamson was poetic, then he executed some quick thinking to run out Michael Bracewell, who was dawdling back for a third run and forgot to ground his bat when Stokes hurled the ball from the boundary.
Leach had to shoulder the burden of sending down 61.3 overs, but got the reward of his fifth Test five-wicket haul.
Southee skied to point, Matt Henry edged to slip and Blundell's swipe finally signalled the start of the England chase.
'We'll take the chase on and entertain'
England's Jack Leach, speaking to BT Sport after the close of play: "It's taken a while to get this point and tomorrow is looking like an exciting day.
"I found it tough at times bowling long spells on these pitches. I just kept going and trusting something would happen. I got a few wickets at the end which was nice.
"Ben Stokes kept me going. He's very supportive and all the guys helped me out.
"It was nice to see Harry Brook get his first Test wicket! I think we tried to do something they were not expecting. If we can throw something a bit different at them we'll do it, and it worked.
"We know how we will go about things to get the win. We will try to chase it. We will be positive, take the game on and try to entertain the crowd."