Walk Around Neighborhood

AtCoder

IDagc062_d

Time4000ms

Memory256MB

Difficulty—

Takahashi, with a notepad, is standing at the origin $(0,0)$ of a two-dimensional plane. His notepad has $N$ **even numbers** $D_i\ (1\leq i \leq N)$ written. Takahashi will perform the following actions $N$ times on the plane. * He chooses and erases one even number written on the notepad. Let $d$ be the chosen even number. He then moves to a point exactly $d$ distance away in terms of Manhattan distance. More precisely, if Takahashi is currently at the coordinates $(x,y)$, he moves to a point $(x',y')$ such that $|x-x'|+|y-y'|=d$. After performing $N$ actions, Takahashi must return to the origin $(0,0)$. Determine if it is possible to perform $N$ actions in such a way. If it is possible, find the minimum value of $\displaystyle \max_{1\leq i \leq N}(|x_i|+|y_i|)$, where $(x_i,y_i)$ are the coordinates where Takahashi is located after the $i$\-th action (we can prove that this value is an integer). ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $2 \leq D_i \leq 2 \times 10^5$ * $D_i\ (1\leq i \leq N)$ is even. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $D_1$ $D_2$ $\dots$ $D_N$ [samples]

Samples

Input #1

3
2 4 6

Output #1

4

If Takahashi erases $2,6,4$ from his notepad for the first to third actions, respectively, and moves as $(0,0)\rightarrow (0,2) \rightarrow (-4,0) \rightarrow (0,0)$, then $\displaystyle \max_{1\leq i \leq N}(|x_i|+|y_i|)$ is $4$, which is the minimum.

Input #2

5
2 2 2 2 6

Output #2

3

If Takahashi erases $2,2,6,2,2$ from his notepad for the first to fifth actions, respectively, and moves as $(0,0)\rightarrow (\frac{1}{2},\frac{3}{2})\rightarrow (0,3) \rightarrow (0,-3) \rightarrow (-\frac{1}{2},-\frac{3}{2}) \rightarrow (0,0)$, then $\displaystyle \max_{1\leq i \leq N}(|x_i|+|y_i|)$ is $3$, which is the minimum.
Takahashi can also move to non-grid points.

Input #3

2
2 200000

Output #3

\-1

Takahashi cannot return to the origin after performing $N$ actions as described above.

API Response (JSON)

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  "problem": {
    "name": "Walk Around Neighborhood",
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      "content": "Takahashi, with a notepad, is standing at the origin $(0,0)$ of a two-dimensional plane. His notepad has $N$ **even numbers** $D_i\\ (1\\leq i \\leq N)$ written. Takahashi will perform the following acti",
      "description_type": "Markdown"
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    "platform": "AtCoder",
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      "time_limit": 4000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "sign": "agc062_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi, with a notepad, is standing at the origin $(0,0)$ of a two-dimensional plane. His notepad has $N$ **even numbers** $D_i\\ (1\\leq i \\leq N)$ written.\nTakahashi will perform the following acti...",
      "is_translate": false,
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