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Problem of the week

Two players play a game with a polynomial with undetermined coefficients \[ 1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8. \] Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.