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code_segments/segment_165.txt

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+This is the hard version of the problem. The only difference is that in this version $k \le 10^{12}$. You can make hacks only if both versions of the problem are solved.
+
+Given a $w \times h$ rectangle on the $Oxy$ plane, with points $(0, 0)$ at the bottom-left and $(w, h)$ at the top-right of the rectangle.
+
+You also have a robot initially at point $(0, 0)$ and a script $s$ of $n$ characters. Each character is either L, R, U, or D, which tells the robot to move left, right, up, or down respectively.
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+The robot can only move inside the rectangle; otherwise, it will change the script $s$ as follows:
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+  * If it tries to move outside a vertical border, it changes all L characters to R's (and vice versa, all R's to L's).    * If it tries to move outside a horizontal border, it changes all U characters to D's (and vice versa, all D's to U's). 
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+Then, it will execute the changed script starting from the character which it couldn't execute.
+
+![](CDN_BASE_URL/ff5ae9758c965c2d8398c936e9581dab) An example of the robot's movement process, $s = \texttt{"ULULURD"}$
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+The script $s$ will be executed for $k$ times continuously. All changes to the string $s$ will be retained even when it is repeated. During this process, how many times will the robot move to the point $(0, 0)$ in total? Note that the initial position does NOT count.
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+The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases.
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+The first line of each test case contains four integers $n$, $k$, $w$, and $h$ ($1 \le n, w, h \le 10^6$; $1 \le k \le 10^{12}$).
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+The second line contains a single string $s$ of size $n$ ($s_i \in \\{\texttt{L}, \texttt{R}, \texttt{U}, \texttt{D}\\}$) — the script to be executed.
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+It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$.
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+For each test case, print a single integer — the number of times the robot reaches $(0, 0)$ when executing script $s$ for $k$ times continuously.
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+In the first test case, the robot only moves up and right for the first two executions. After th