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For each $n\\geq 1$ and any integer $d\\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\\mathbb{Z}^n})$ such that (1) for any distinct $x,y\\in M$ in the same orbit, $\\rho(x,y)\\geq d$; (2) for any $1\\leq i\\leq n$ and any $x\\in F(2^{\\mathbb{Z}^n})$, there are non-negative integers $a, b\\leq D$ such that $a\\cdot x\\in M$ and $-b\\cdot x\\in M$. As an application, we obtain a clopen tree section for $F(2^{\\mathbb{Z}^n})$. 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For each $n\\geq 1$ and any integer $d\\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\\mathbb{Z}^n})$ such that (1) for any distinct $x,y\\in M$ in the same orbit, $\\rho(x,y)\\geq d$; (2) for any $1\\leq i\\leq n$ and any $x\\in F(2^{\\mathbb{Z}^n})$, there are non-negative integers $a, b\\leq D$ such that $a\\cdot x\\in M$ and $-b\\cdot x\\in M$. As an application, we obtain a clopen tree section for $F(2^{\\mathbb{Z}^n})$. 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