Extending $(0,1)$ by one even target $t\equiv2\pmod4$ with $\varepsilon_0=0$ forces $t=2\cdot3^n$ and odd letters $=\{3^i\}$

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Extending $(0,1)$ by one even target $t\equiv2\pmod4$ with $\varepsilon_0=0$ forces $t=2\cdot3^n$ and odd letters $=\{3^i\}$ · Prove2Me