According to what Genya taught Tanjiro, 'Repetitive Actions' is a technique that opens all five senses. Genya may not be able to use breaths, but he can achieve full focus another way through repetitive actions. When Gyomei and Genya use it, they remember their anger and painful memories. That makes their heart rate and temperature go up. The signs were pointing to the possibility that this might be the same as Tanjiro's mark coming out, but Gyomei and Genya had no marks, so they cocked their head in confusion.
By doing these repetitive actions, you can focus at any given moment and Tanjiro thought how it would be nice if he could learn how to maintain his darkened mark forever. His repetitive actions were remembering all the faces of his loved ones. Then, recalling Rengoku's words “set you heart ablaze!” and with that, he was able to raise his focus to its maximum limit. Tanjiro couldn't do it in the beginning but he was able to use all the strength he had by doing these repetitive actions. As he did them again and again, his body began to remember the way to bring out that power through these actions. Eventually, he was able to push the boulder, slowly but surely. Inosuke was also able to push his by thinking about tempura with Zenitsu looking dismayed as he will be left out only to be distracted by his sparrow with a letter addressed to him.
Tanjiro was able to move his boulder across a town finishing the Rock Hashira's training but was left dehydrated. Gyomei then gave him water and accepted him as a swordsman recalling Tanjiro's actions in the swordsmith village when he prioritized the villagers over his younger sister during the crisis. Tanjiro however did not accept his praise as Nezuko was the one who made that decision and not him. Upon hearing this, Gyomei noted how different Tanjiro is as to him, children are pure and naive, weak, quick to lie, completely fine with cruelty, and selfish little things.
Gyomei then told Tanjiro his story about how he raised some kids who had no family in a temple before he became a hunter. Their affinity in having no blood connections helped them bond together very well and he raised them like his family, Gyomei wanted to keep life that way forever. But then one night, a kid disobeyed the temple rules and didn't come back after sunset and ran into a demon. In order to save himself, the boy told the demon to eat Gyomei and the 8 kids at the temple. The kid put out all the wisteria incense Gyomei burned as protection and invited the demon into the temple which resulted in 4 kids dying immediately. Gyomei tried to protect the other 4 but 3 didn't listen to him, ran and died, only Sayo (the youngest) listened and hid behind.
In order to protect Sayo, Gyomei fought and punched the demon's head again and again until the sun came up. He was able to protect Sayo but when people came to help, Sayo who was only 4 years old that time got confused and labelled Gyomei a monster who killed all the children. As a result, Gyomei was imprisoned and charged with murder. If Kagaya had not intervened, he would have been executed and ever since, he had come to harbor deep doubts and suspicions that no matter how virtuous you are, once you get pushed into a corner, your true nature will come out. But Tanjiro never ran and stayed honest and never lied. Seeing as Tanjiro is a special child, Gyomei promised to help him so that Tanjiro won't take a wrong turn in life.