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For log₂(log₃(log_0.5(cos(x) +a))) to exist

log₃(log_0.5(cos(x) +a)) > 0

(log_0.5(cos(x) +a) > 3⁰ = 1

cosx + a < (0.5)¹ (since the base of above log is less than 1).

a < 0.5 - cosx

cosx can attain a maximum value of 1.

Hence a < -0.5....................(1)

Also for (log_0.5(cos(x) +a) to be defined

cosx + a > 0

a > -cosx

cosx can attain a minimum value of -1.

Hence a > 1...................(2)

Inequalities (1) and (2) cannot exist simultaneously

Hence a has no possible real value.

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