A Game with Traps
Contest: Educational Codeforces Round 77 (Rated for Div. 2)
Difficulty rating: 1900
Tags: binary search, dp, greedy, sortings

You are playing a computer game, where you lead a party of $$$m$$$ soldiers. Each soldier is characterised by his agility $$$a_i$$$.

The level you are trying to get through can be represented as a straight line segment from point $$$0$$$ (where you and your squad is initially located) to point $$$n + 1$$$ (where the boss is located).

The level is filled with $$$k$$$ traps. Each trap is represented by three numbers $$$l_i$$$, $$$r_i$$$ and $$$d_i$$$. $$$l_i$$$ is the location of the trap, and $$$d_i$$$ is the danger level of the trap: whenever a soldier with agility lower than $$$d_i$$$ steps on a trap (that is, moves to the point $$$l_i$$$), he gets instantly killed. Fortunately, you can disarm traps: if you move to the point $$$r_i$$$, you disarm this trap, and it no longer poses any danger to your soldiers. Traps don't affect you, only your soldiers.

You have $$$t$$$ seconds to complete the level — that is, to bring some soldiers from your squad to the boss. Before the level starts, you choose which soldiers will be coming with you, and which soldiers won't be. After that, you have to bring all of the chosen soldiers to the boss. To do so, you may perform the following actions:

- if your location is $$$x$$$, you may move to $$$x + 1$$$ or $$$x - 1$$$. This action consumes one second;
- if your location is $$$x$$$ and the location of your squad is $$$x$$$, you may move to $$$x + 1$$$ or to $$$x - 1$$$ with your squad in one second. You may not perform this action if it puts some soldier in danger (i. e. the point your squad is moving into contains a non-disarmed trap with $$$d_i$$$ greater than agility of some soldier from the squad). This action consumes one second;
- if your location is $$$x$$$ and there is a trap $$$i$$$ with $$$r_i = x$$$, you may disarm this trap. This action is done instantly (it consumes no time).

Note that after each action both your coordinate and the coordinate of your squad should be integers.

You have to choose the maximum number of soldiers such that they all can be brought from the point $$$0$$$ to the point $$$n + 1$$$ (where the boss waits) in no more than $$$t$$$ seconds.

Input format:
The first line contains four integers $$$m$$$, $$$n$$$, $$$k$$$ and $$$t$$$ ($$$1 \le m, n, k, t \le 2 \cdot 10^5$$$, $$$n < t$$$) — the number of soldiers, the number of integer points between the squad and the boss, the number of traps and the maximum number of seconds you may spend to bring the squad to the boss, respectively.

The second line contains $$$m$$$ integers $$$a_1$$$, $$$a_2$$$, ..., $$$a_m$$$ ($$$1 \le a_i \le 2 \cdot 10^5$$$), where $$$a_i$$$ is the agility of the $$$i$$$-th soldier.

Then $$$k$$$ lines follow, containing the descriptions of traps. Each line contains three numbers $$$l_i$$$, $$$r_i$$$ and $$$d_i$$$ ($$$1 \le l_i \le r_i \le n$$$, $$$1 \le d_i \le 2 \cdot 10^5$$$) — the location of the trap, the location where the trap can be disarmed, and its danger level, respectively.

Output format:
Print one integer — the maximum number of soldiers you may choose so that you may bring them all to the boss in no more than $$$t$$$ seconds.

Constraints:
Time limit: 3.0 s
Memory limit: 256.0 MB

Note:
In the first example you may take soldiers with agility $$$3$$$, $$$4$$$ and $$$5$$$ with you. The course of action is as follows:

- go to $$$2$$$ without your squad;
- disarm the trap $$$2$$$;
- go to $$$3$$$ without your squad;
- disartm the trap $$$3$$$;
- go to $$$0$$$ without your squad;
- go to $$$7$$$ with your squad.

The whole plan can be executed in $$$13$$$ seconds.

Example input:
5 6 4 14
1 2 3 4 5
1 5 2
1 2 5
2 3 5
3 5 3

Example output:
3

Editorial:
When we fix a set of soldiers, we can determine a set of traps that may affect our squad: these are the traps with danger level greater than the lowest agility value. So we can use binary search on minimum possible agility of a soldier that we can choose.
How should we actually bring our soldiers to the boss? Each trap that can affect our squad can be actually treated as a segment $$$[l_i, r_i]$$$ such that our squad cannot move to $$$l_i$$$ until we move to $$$r_i$$$ and disarm this trap. We should walk through such segments for three times: the first time we walk forwards without our squad to disarm the trap, the second time we walk backwards to return to our squad, and the third time we walk forwards with our squad. So the total time we have to spend can be calculated as $$$n + 1 + 2T$$$, where $$$T$$$ is the number of unit segments belonging to at least one trap-segment — and it can be calculated with event processing algorithms or with segment union.
Time complexity is $$$O(n \log n)$$$ or $$$O(n \log^2 n)$$$, but it is possible to write a solution in $$$O(n \alpha(n))$$$ without binary search.