TOKYO — Sanae Takaichi, a hard-line conservative lawmaker, won a critical leadership vote Saturday, putting her on track to become Japan’s first female prime minister, a milestone in a country where women are vastly underrepresented in politics.

Takaichi, 64, prevailed in an election by the governing Liberal Democratic Party after two rounds of voting by lawmakers and rank-and-file members. Her victory could herald a rightward shift in Japan, where antiestablishment politicians have recently made gains with voters concerned about stagnant wages, rising prices and an influx of foreign workers and tourists.

Takaichi, in a short speech after her victory before a crowd of LDP leaders gathered at party headquarters in Tokyo, promised to “work, work and work” and exhorted her colleagues to “wo

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