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code_segments/segment_373.txt

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+For an arbitrary binary string $t$$^{\text{∗}}$, let $f(t)$ be the number of non-empty subsequences$^{\text{†}}$ of $t$ that contain only $\mathtt{0}$, and let $g(t)$ be the number of non-empty subsequences of $t$ that contain at least one $\mathtt{1}$.
+
+Note that for $f(t)$ and for $g(t)$, each subsequence is counted as many times as it appears in $t$. E.g., $f(\mathtt{000}) = 7, g(\mathtt{100}) = 4$.
+
+We define the oneness of the binary string $t$ to be $|f(t)-g(t)|$, where for an arbitrary integer $z$, $|z|$ represents the absolute value of $z$.
+
+You are given a positive integer $n$. Find a binary string $s$ of length $n$ such that its oneness is as small as possible. If there are multiple strings, you can print any of them.
+
+$^{\text{∗}}$A binary string is a string that only consists of characters $\texttt{0}$ and $\texttt{1}$.
+
+$^{\text{†}}$A sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements. For example, subsequences of $\mathtt{1011101}$ are $\mathtt{0}$, $\mathtt{1}$, $\mathtt{11111}$, $\mathtt{0111}$, but not $\mathtt{000}$ nor $\mathtt{11100}$.
+
+The first line contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases.
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+The only line of each test case contains an integer $n$ ($1 \leq n \leq 2\cdot10^5$) — the length of $s$.
+
+It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot10^5$.
+
+For each test case, output $s$ on a new line. If multiple answers exist, output any.
+
+In the first test case, for the example output, $f(t)=1$ because there is one subsequence that contains only $\mathtt{0}$ ($\mathtt{0}$), and $g(t)=0$ because there are no subsequences that contain at least one $1$. The oneness is $|1-0|=1$. The output $\mathtt{1}$ is correct as well because its oneness is $|0-1|=1$.
+
+For the example output of the second test case, $f(t)=1$ because there is one non-empty subsequence that contains only $\mathtt{0}$, and $g(t)=2$ because there are two non-empty subs