The number of such points \( (a+1, \sqrt{3} a) \), where \( a \) is...

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The number of such points \( (a+1, \sqrt{3} a) \), where \( a \) is any integer, lying inside the region bounded by the circles
\( \mathrm{P} \) \( x^{2}+y^{2}-2 x-3=0 \) and \( x^{2}+y^{2}-2 x-15=0 \), is

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