Question 120
The sequence $\{a_n\}$ is defined as follows: $a_1 = 0$, and for integer $n \ge 2$,
$$
a_n = \frac{1}{n} + \frac{1}{\lceil \frac{n}{2} \rceil} \sum_{k=1}^{\lceil \frac{n}{2} \rceil} a_k,
$$
where $\lceil \frac{n}{2} \rceil$ denotes the smallest integer not less than $\frac{n}{2}$. Find the maximum term of the sequence $\{a_n\}$.
$$
a_n = \frac{1}{n} + \frac{1}{\lceil \frac{n}{2} \rceil} \sum_{k=1}^{\lceil \frac{n}{2} \rceil} a_k,
$$
where $\lceil \frac{n}{2} \rceil$ denotes the smallest integer not less than $\frac{n}{2}$. Find the maximum term of the sequence $\{a_n\}$.