Change directory manager

Parse `cd` commands and compute the absolute path after a series of directory changes.

string parsing paths stack

You are given a starting absolute path (e.g., `"/a/b"`) and a list of `cd` commands (e.g., `["..", "c", "d/", "."]`). Your task is to simulate the commands and return the resulting absolute path.

Define the function `cd_manager(start: str, commands: list[str]) -> str` that returns the absolute path after applying all commands.

Rules:
- The starting path is always absolute (begins with `/`) and may contain multiple slashes (e.g., `"/a//b/"`). You should normalize it to a single slash between components.
- Each command is a string. It can be:
  - `".."` — move up one directory. If already at root (`"/"`), stay at root.
  - `"."` — stay in the current directory (no effect).
  - a directory name (possibly with trailing slashes, e.g., `"c/"`) — move into that directory.
- Commands are applied sequentially.
- The output must be a normalized absolute path: no trailing slashes except for root `"/"`, and no consecutive slashes.
- You may also have commands like `"folder/"` (trailing slash), which should be treated as `"folder"`.

Implement the function without using any path manipulation libraries (like `os.path`). Use string processing and a stack.

Note: The input will always be valid; you do not need to handle invalid commands.

Constraints

- `1 <= len(start) <= 200`
- `0 <= len(commands) <= 200`
- Each command is either `".."`, `"."`, or a non-empty string of lowercase letters and digits, possibly with one or more trailing slashes.
- Total length of all commands combined <= 2000.
- No commands contain `"/"` inside except trailing slashes.
- Time complexity: O(total length of start + commands).
- Space: O(number of components).

Example

```python
>>> cd_manager("/a/b", ["..", "c", "d/", "."])
"/a/c/d"
>>> cd_manager("/a//b/", ["..", "..", ".."])
"/"
>>> cd_manager("/", ["usr", "local", ".."])
"/usr"
```

20 points ~20 min

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Hints

Split the start path by `/` and ignore empty strings to get components.

Use a stack to keep track of the current directory components; pop on `..` and push on directory names.

Remember to normalize the final path: join with `/` and prepend `/`; if stack is empty, return `"/"`.

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