Calculate Average Grades and Sort Students

You are given a JSON array representing a list of students. Each student is an object with a name (a string) and grades (an array of integers).

Your task is to:

• Compute the average grade for each student using the formula: $$\text{average} = \frac{\sum_{i=1}^{n} grade_i}{n}$$. Round the result to two decimal places.
• Add the computed average to the corresponding student object with the key average.
• Sort the list of students in descending order by average grade.
• Output the sorted list in JSON format.

Note: The input is provided as a JSON formatted string via standard input and the output should be printed to standard output.

inputFormat

The input consists of a single JSON array provided via standard input. Each element in the array is a JSON object with the following structure:

{
  "name": "Student Name",
  "grades": [grade1, grade2, ..., gradeN]
}

There are no extra lines in the input.

outputFormat

The output should be a JSON array printed to standard output. Each object in the array must contain the original name and grades, and an additional average key representing the student's average grade (rounded to two decimal places). The array must be sorted in descending order by the average value.

## sample

[
  {"name": "John Doe", "grades": [88, 92, 79]},
  {"name": "Jane Smith", "grades": [85, 90, 95]},
  {"name": "Emily Davis", "grades": [70, 75, 80]}
]

[
{
    "name": "Jane Smith",
    "grades": [85, 90, 95],
    "average": 90.0
},
{
    "name": "John Doe",
    "grades": [88, 92, 79],
    "average": 86.33
},
{
    "name": "Emily Davis",
    "grades": [70, 75, 80],
    "average": 75.0
}

]

</p>