Wales looked set to end their 13-match losing streak with a dramatically improved performance in Cardiff.
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Wales led from prop Rhys Carré’s ninth-minute try to George Turner’s winning try for Scotland six minutes from time.
“It’s heartbreaking. It’s a hard feeling because you know the performance was there,” Lake said.
“We have delivered the things we talked about during the week. In the first half we dominated, pressed and attacked.
“We talked a lot about staying in the moments and we didn’t do it during the 80 minutes, that’s why the scoreboard was against us.
“We probably did enough to win the game, which makes it so difficult.”
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Tandy enjoyed a victory in the fall against Japan, but came agonizingly close to a first victory of his reign.
Wales led at half-time for the first time in charge and were 20-5 and 23-12 ahead at times in the second half. But in the end they only lost one bonus point.
“I’m very disappointed that we didn’t win the game, but I’m very proud of the group,” Tandy said.
“That was a step in the right direction, but we have to be better next time. There were a lot of positives in the first half with the way we attacked the game. That was excellent.”
After French fans invaded Principality Stadium in the second round, Tandy praised the impact of home support from a crowd of 70,649 inside Principality Stadium.
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“The support today was incredible. A lot of questions were asked about the crowd, but they were amazing,” Tandy said.
“Everyone felt it. The players gave them something to be happy about, but in the end they supported the team. It’s a snapshot of what we want to be.”
Scarlets pair Taine Plumtree and Sam Costelow appear unlikely to play in the tournament again.
Defender Plumtree (shoulder) and flyhalf Costelow (lower leg) will undergo tests on Monday.
“Sam doesn’t look good,” Tandy said.
“He has been through a lot in his career in Wales without being involved in the autumn, but we backed him to come back and he really took advantage of that opportunity.
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“He was fantastic with the way he drove the team forward in the first half. It’s a shame for him that it all ended with an injury.”
Wales ‘tactically exceptional’
Tandy was Scotland’s defense coach for six years and his game plan earned praise from his former boss.
“Tactically they were outstanding,” Gregor Townsend said.
“Wales moved the ball quickly, they played around the ruck which was difficult to defend, their set piece was excellent and defensively, something we probably hadn’t seen from them. They targeted our attack and kept their width.
“They prepared the team very well and they have excellent players; we knew that Wales would play well and we knew the quality they have in the team.
“They deserved to be ahead and it took a great effort to get back into the game and get ahead on the scoreboard.”