Let $a_1, a_2, \dots, a_n$ be a beautiful sequence of length $n$. Take an integer $m$ such that $a_m = 2025$. Then $m \ge 2$ since $a_1 = 0$.
We first prove $n \le 13$. Let $t$ be an integer with $1 \le t \le n - 2$. By applying the third condition to $(i, j, k) = (t, t+1, n)$, we obtain
$$\frac{a_t + a_n}{2} \le a_{t+1},$$
which implies
$$a_n - a_t \ge 2(a_n - a_{t+1}).$$
If $m = n$, repeatedly applying this inequality gives
$$2025 = a_n - a_1 \ge 2(a_n - a_2) \ge \dots \ge 2^{n-2}(a_n - a_{n-1}) \ge 2^{n-2},$$
and hence $n \le 12$. If $m < n$, we have
$$a_n - a_1 \ge 2(a_n - a_2) \ge \dots \ge 2^{m-2}(a_n - a_{m-1}) \ge 2^{m-1}(a_n - a_m),$$
and
$$a_n - a_m \ge 2(a_n - a_{m+1}) \ge 2^2(a_n - a_{m+2}) \ge \dots \ge 2^{n-m-1}(a_n - a_{n-1}) \ge 2^{n-m-1}.$$
$$\begin{aligned} 2025 &= a_m - a_1 = (a_n - a_1) - (a_n - a_m) \ge (2^{m-1} - 1)(a_n - a_m) \\ &\ge (2^{m-1} - 1)2^{n-m-1} \ge 2^{m-2} \cdot 2^{n-m-1} = 2^{n-3} \end{aligned}$$
$\quad$ (*)

and so we obtain $n \le 13$.
Let $n = 13$. Then the above argument shows $n > m$. From $(*)$, we have

2025 $\ge (2^{m-1} - 1)2^{n-m-1} = 2^{n-2} - 2^{n-m-1} = 2^{11} - 2^{12-m},$
$$which implies \$2^{12-m} \ge 23\$, and hence \$2^{12-m} \ge 32\$. Therefore,$$
a_n = a_m + (a_n - a_m) $\ge$ a_m + $2^{n-m-1}$ = 2025 + $2^{12-m} \ge$ 2057.
$$Thus, we have \$a_n \ge 2057\$ when \$n = 13\$. Finally, we show that the sequence$$
(a_1, a_2, $\dots, a_{13})$ = (0, 1033, 1545, 1801, 1929, 1993, 2025, 2041, 2049, 2053, 2055, 2056, 2057)

is a beautiful sequence of length 13. The first and second conditions are clearly satisfied. We now verify the third condition. We have

$(a_{13}$ - a_1, $a_{13}$ - a_2, $\dots, a_{13} - a_{12})$ = (2057, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1),
$$so for all integers \$t\$ with \$1 \le t \le 11\$, we have \$a_{13} - a_t \ge 2(a_{13} - a_{t+1})\$. Therefore, for all integers \$1 \le i < j < k \le 13\$, we have$$
$$\begin{aligned} a_j - \frac{a_i + a_k}{2} &\ge a_j - \frac{a_i + a_{13}}{2} = \frac{(a_{13} - a_i) - 2(a_{13} - a_j)}{2} \\ &\ge \frac{(a_{13} - a_i) - (a_{13} - a_{j-1})}{2} = \frac{a_{j-1} - a_i}{2} \\ &\ge 0. \end{aligned}$$
$$$$
This confirms the third condition.
Therefore, the maximum value of the length of a beautiful sequence is 13 and for beautiful sequences $a_1, a_2, \dots, a_{13}$ of length 13, the minimum value of $a_{13}$ is 2057.