Packing Biscuits

Aunty Khong is organising a competition with $x$ participants, and wants to give each participant a bag of biscuits. There are $k$ different types of biscuits, numbered from $0$ to $k-1$. Each biscuit of type $i$ ($0 \leq i \leq k-1$) has a tastiness value of $2^i$. Aunty Khong has $a\[i]$ (possibly zero) biscuits of type $i$ in her pantry.

Each of Aunty Khong's bags will contain zero or more biscuits of each type. The total number of biscuits of type $i$ in all the bags must not exceed $a\[i]$. The sum of tastiness values of all biscuits in a bag is called the total tastiness of the bag.

Help Aunty Khong find out how many different values of $y$ exist, such that it is possible to pack $x$ bags of biscuits, each having total tastiness equal to $y$.

• $1 \leq k \leq 60$
• $1 \leq q \leq 1000$
• $1 \leq x \leq 10^{18}$
• $0 \leq a\[i] \leq 10^{18}$ (for all $0 \leq i \leq k-1$)
• For each call to count_tastiness, the sum of tastiness values of all biscuits in the pantry does not exceed $10^{18}$.