Parse a Signed Integer in an Arbitrary Base

Implement a production-style conversion from a signed string to a 32-bit integer in a supplied base. Implement `parse_base_integer(text: string, base: int) -> string`. Return the canonical base-10 integer on success. Return `INVALID` for invalid syntax or an unsupported base, and `OVERFLOW` for a syntactically valid value outside signed 32-bit range. ### Constraints & Assumptions The source requests arbitrary-base atoi and explicitly notes the negative-number case. The following strict parsing and error policy is an explicit practice contract. - `2 <= base <= 36` is supported; any other base returns `INVALID`. - Input length is at most 100000 characters. Trim only ASCII spaces at the two ends. Internal whitespace, tabs, separators, and trailing non-digit text are invalid. No base prefix is specially recognized or stripped: all characters after the optional sign are interpreted by the digit grammar for the supplied base. Thus `0x1` is valid ordinary digits in base 36 (value 1189), but invalid in base 16 because `x` is not a base-16 digit. - After trimming, accept an optional single `+` or `-`, followed by at least one digit. Digits are `0`–`9`, `a`–`z`, or `A`–`Z`, with letters representing 10 through 35, case-insensitively. Every digit value must be less than base. - The signed range is -2147483648 through 2147483647. Leading zeroes are allowed. Both `-0` and `+0` return `0`. - Syntax takes precedence over overflow: if any character violates the grammar, return `INVALID` even if an earlier prefix already overflowed. Otherwise return `OVERFLOW` if the magnitude is too large. - Aim for O(length) time and O(1) auxiliary space. Do not depend on arbitrary-precision conversion of the whole input. ### Examples ```text parse_base_integer(" -80000000 ",16) -> "-2147483648" parse_base_integer("7fffffff",16) -> "2147483647" parse_base_integer("80000000",16) -> "OVERFLOW" parse_base_integer("10102",2) -> "INVALID" parse_base_integer("+000",10) -> "0" parse_base_integer("0x1",36) -> "1189" parse_base_integer("0x1",16) -> "INVALID" ``` Explain validation of empty/sign-only strings and how the negative boundary differs from the positive boundary. Describe how you detect overflow before multiplication and addition. ```hint Check against a sign-specific magnitude bound Before appending digit d to accumulated magnitude v, compare v with the largest value for which `v * base + d` remains allowed. Continue syntax validation after detecting overflow. ```