Fewest inserts to balance brackets

Question

Report how many bracket characters must be inserted anywhere in a parenthesis string to make every prefix and the whole line balanced.

```python
def solve(raw: str) -> str: ...
```

## Requirements

- The input is one line containing only `(` and `)`.
- A string is balanced when every prefix has at least as many `(` as `)`, and the total counts are equal.
- Insertions may be placed at any position. Removing, replacing, or reordering characters is not allowed.
- Return the minimum number of inserted characters as a decimal string.
- A line that is already balanced returns `"0"`.
- `run_case(case: dict)` forwards `case["stdin"]` to `solve`.
- Do not append a trailing newline to the returned string; return the value exactly once.