Two children collaborate on a problem for amusement. They place three coins, all of which are unit circles, inside a circular playing field of radius \(R\).

They take turns making a move, with the following provisions

**1)** One coin is moved in each turn, not necessarily (but may be) in a straight path, sliding it from where it is to a new position inside the circular playing field without disturbing the other two coins,

**2)** such that the center of the moved coin slides past the line connecting the centers of the other two coins (i.e., sliding between the other two coins) **exactly once**,

**3)** and to a new position not necessarily having to (but may) touch any of the other two coins or the border of the circular playing field.

**4)** The same coin may not be moved in two consecutive turns.

**5)** The two children deliberately play in such a way to make possible taking turns indefinitely, i.e., not being forced to stop playing because it has become impossible to move a coin and have its center pass through the line connecting centers of the other two coins without disturbing them.

**6)** The two children deliberately at the start place the three coins in the circular playing field in such a way that such indefinitely extended play is made possible.

What is the infimum **radius** \(R\) of the circular playing field for this to be possible?

Submit your answer as \(\left\lfloor 1000 R \right\rfloor \).

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