what can I use as a synonym for "take the lead" preferably in one word?
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The most straightforward is simply "lead": "you lead, I'll follow" is equivalent to "you take the lead, I'll follow". It works for "lead" in the sense of tasks or actual paths.
There are plenty of synonymous phrases but all the single word synonyms (I can think of) only work in some situations, and have other connotations anyway ("guide", "direct").
I think "spearhead" is a good option.
serve as leader or leading element of
Source: Merriam Webster