Palindrome after at most k deletions

Question

Given a lowercase string and a deletion limit, decide whether removing at most that many characters can leave a palindrome while the surviving characters keep their order.

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


def run_case(case: dict) -> str:
    return solve(case["stdin"])
```

## Requirements

- Line 1 is the string; line 2 is the integer limit.
- Removing nothing is allowed, so a string that already reads the same in both directions always qualifies.
- Each deletion is a single character, and characters that are kept never change position relative to each other.
- On a mismatched pair, only two moves are worth trying: drop the left character or drop the right one.
- Return `true` or `false` in lowercase, with no other text and no extra whitespace.
- The string can be long, so do not try every subset of deletions; reuse the two-way choice.
- Reporting which characters were removed is out of scope.
- Do not append a trailing newline to the returned string; return the value exactly once.