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code_segments/segment_229.txt

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+This is the hard version of the problem. In this version, it is not guaranteed that $n=m$, and the time limit is higher. You can make hacks only if both versions of the problem are solved.
+
+In the court of the Blue King, Lelle and Flamm are having a performance match. The match consists of several rounds. In each round, either Lelle or Flamm wins.
+
+Let $W_L$ and $W_F$ denote the number of wins of Lelle and Flamm, respectively. The Blue King considers a match to be successful if and only if:
+
+  * after every round, $\gcd(W_L,W_F)\le 1$;    * at the end of the match, $W_L\le n, W_F\le m$. 
+
+Note that $\gcd(0,x)=\gcd(x,0)=x$ for every non-negative integer $x$.
+
+Lelle and Flamm can decide to stop the match whenever they want, and the final score of the performance is $l \cdot W_L + f \cdot W_F$.
+
+Please help Lelle and Flamm coordinate their wins and losses such that the performance is successful, and the total score of the performance is maximized.
+
+The first line contains an integer $t$ ($1\leq t \leq 10^3$) — the number of test cases.
+
+The only line of each test case contains four integers $n$, $m$, $l$, $f$ ($2\leq n\leq m \leq 2\cdot 10^7$, $1\leq l,f \leq 10^9$): $n$, $m$ give the upper bound on the number of Lelle and Flamm's wins, $l$ and $f$ determine the final score of the performance.
+
+Unusual additional constraint: it is guaranteed that, for each test, there are no pairs of test cases with the same pair of $n$, $m$.
+
+For each test case, output a single integer — the maximum total score of a successful performance.
+
+In the first test case, a possible performance is as follows:
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+  * Flamm wins, $\gcd(0,1)=1$.    * Lelle wins, $\gcd(1,1)=1$.    * Flamm wins, $\gcd(1,2)=1$.    * Flamm wins, $\gcd(1,3)=1$.    * Flamm wins, $\gcd(1,4)=1$.    * Lelle and Flamm agree to stop the match. 
+
+The final score is $1\cdot2+4\cdot5=22$.