Key observation in this problem is that gcd(x₁, ..., x_n) = gcd(x₁, x₂ - x₁, x₃ - x₂, ..., x_n - x_n - 1). It is quite easy to understand : if all x_i are divisible by d, then all x_i + 1 - x_i are divisible by d, if x_i + 1 - x_i are divisible by d, then their prefix sum x₁ + (x₂ - x₁) + ... + (x_i + 1 - x_i) = x_i + 1 is divisible by d.

First, you can keep a_i in segment tree with operations "add arithmetic progression on segment" and "request i-th element". For simplicity, I'll assume r > l + 3 later, other cases are solved easily when we have such segment tree.

Second, gcd(a_l, ..., a_r) = gcd(a_l, a_l + 1 - a_l, a_l + 2 - a_l + 1, ..., a_r - a_r - 1) = gcd(a_l, gcd(a_l + 1 - a_l, a_l + 2 - a_l + 1, ..., a_r - a_r - 1)).

Let's say b_i + 1 = a_i + 1 - a_i. Let's forget about existence of a for a moment and reformulate statement in new variables: in first operation we need to calculate gcd on segment, thanks to above equality, in second operation we need to add k to some segment of b, and add  - (r - l + 1)k to some other position of b. So we can use segment tree on b_i in similar fashion to find b_l + 1 = a_l + 1 - a_l. But gcd(b_l + 1, ..., b_r) = gcd(b_l + 1, gcd(b_l + 2 - b_l + 1, ..., b_r - b_r - 1). So if c_i + 1 = b_i + 1 - b_i, then we need to calculate gcd(c_l + 2, ..., c_r). But when we add some value to segment of b not more than two values of c change, so we can use usual segment tree for "change single value" and "calculate gcd on segment" for c.

The answer is gcd(gcd(a_l, b_l + 1), gcd(c_l + 2, ..., c_r)), which can be calculated using described segment trees easily.