Let  0<a<b.  Evaluate $$\lim_{t \to 0} \left \{ {\int_{0}^{1} [bx+a(1-x)]^{t}dx}\right \} ^{1/t}$$[The  final  answer should  not  involve  any  operations other  than  addition,  subtraction,  multiplication,  division,  and  exponentiation.] When \(a = 1\) and \(b = 2\), the limit can be written as \(\frac{a}{be}\), where both \(a\) and \(b\) are positive integers. Find the value of \(a + b\).

Let's think step by step and output the final answer within \boxed{}.
user