(cod: P-80-8-10) Let \(a_1, a_2, \dots, a_n\) be positive numbers and consider the function \[f(t) = \left\{\begin{array}{ll} \left(\dfrac{a_1^t + \cdots + a_n^t}{n}\right)^{\frac{1}{t}}, & \text{ if } t \ne 0 \\ A, & \text{ if } t=0.\end{array} \right. \] Observe that, for each fixed \(t \ne 0\), \(f(t)\) represents a type of mean of the numbers \(a_1, \dots, a_n\). For example, \(f(1)\) is the arithmetic mean of these numbers, while \(f(-1)\) is the harmonic mean of the same numbers. Determine the value of the constant \(A\) so that the function \(f\) is continuous at \(t=0\).