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code_segments/segment_376.txt

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+Suppose we partition the elements of an array $b$ into any number $k$ of non-empty multisets $S_1, S_2, \ldots, S_k$, where $k$ is an arbitrary positive integer. Define the score of $b$ as the maximum value of $\operatorname{MEX}(S_1)$$^{\text{∗}}$$ + \operatorname{MEX}(S_2) + \ldots + \operatorname{MEX}(S_k)$ over all possible partitions of $b$ for any integer $k$.
+
+Envy is given an array $a$ of size $n$. Since he knows that calculating the score of $a$ is too easy for you, he instead asks you to calculate the sum of scores of all $2^n - 1$ non-empty subsequences of $a$.$^{\text{†}}$ Since this answer may be large, please output it modulo $998\,244\,353$.
+
+$^{\text{∗}}$$\operatorname{MEX}$ of a collection of integers $c_1, c_2, \ldots, c_k$ is defined as the smallest non-negative integer $x$ that does not occur in the collection $c$. For example, $\operatorname{MEX}([0,1,2,2]) = 3$ and $\operatorname{MEX}([1,2,2]) = 0$
+
+$^{\text{†}}$A sequence $x$ is a subsequence of a sequence $y$ if $x$ can be obtained from $y$ by deleting several (possibly, zero or all) elements.
+
+The first line contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases.
+
+The first line of each test case contains an integer $n$ ($1 \leq n \leq 2 \cdot 10^5$) — the length of $a$.
+
+The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \leq a_i < n$) — the elements of the array $a$.
+
+It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
+
+For each test case, output the answer, modulo $998\,244\,353$.
+
+In the first testcase, we must consider seven subsequences:
+
+  * $[0]$: The score is $1$.    * $[0]$: The score is $1$.    * $[1]$: The score is $0$.    * $[0,0]$: The score is $2$.    * $[0,1]$: The score is $2$.    * $[0,1]$: The score is $2$.    * $[0,0,1]$: The score is $3$. 
+
+The answer for the first testcase is $1+1+2+2+2+3=11$.
+
+In the last testcase, all subsequences have a score of $0$.