// Time:  O(max(x, max(forbidden)) + a + (b+a))
// Space: O(max(x, max(forbidden)) + a + (b+a))

class Solution {
public:
    int minimumJumps(vector<int>& forbidden, int a, int b, int x) {
        int max_f = *max_element(cbegin(forbidden), cend(forbidden));
        int max_val = (a >= b) ? x + b : max(x, max_f) + a + b;  // a may be a non-periodic area, (a+b) is a periodic area which is divided by gcd(a, b) and all points are reachable
        vector<unordered_set<int>> lookup(2); 
        for (const auto& pos : forbidden) {
            lookup[0].emplace(pos);
            lookup[1].emplace(pos);
        }
        int result = 0;
        vector<pair<int, int>> q = {{0, true}};
        lookup[0].emplace(0);
        while (!empty(q)) {
            vector<pair<int, int>> new_q;
            for (const auto& [pos, can_back] : q) {
                if (pos == x) {
                    return result;
                }
                if (pos + a <= max_val && !lookup[0].count(pos + a)) {
                    lookup[0].emplace(pos + a);
                    new_q.emplace_back(pos + a, true);
                }
                if (!can_back) {
                    continue;
                }
                if (pos - b >= 0 && !lookup[1].count(pos - b)) {
                    lookup[1].emplace(pos - b);
                    new_q.emplace_back(pos - b, false);
                }
            }
            q = move(new_q);
            ++result;
        }
        return -1;
    }
};