your solution shouldn't pass in the first place, maximum time complexity here is 2e5*1e4 = 2e9

I'm at my wit's end. Can anyone else understand what's going on here? I made 5 submissions to this problem with and without cout.tie(NULL) to check that this is not simply a random fluctuation (my working hypothesis) and indeed, with cout.tie(NULL) it gets AC every time (albeit always >960 ms), without it never. Does Codeforces cache some results of submissions that generate identical binaries? Did I not make enough attempts?

Here is the source code of .tie() that comes with winlibs GCC 13.2, which Codeforces uses. It is located at /mingw64/include/c++/13.2.0/bits.

      /**
       *  @brief  Fetches the current @e tied stream.
       *  @return  A pointer to the tied stream, or NULL if the stream is
       *           not tied.
       *
       *  A stream may be @e tied (or synchronized) to a second output
       *  stream.  When this stream performs any I/O, the tied stream is
       *  first flushed.  For example, @c std::cin is tied to @c std::cout.
      */
      basic_ostream<_CharT, _Traits>*
      tie() const
      { return _M_tie; }

      /**
       *  @brief  Ties this stream to an output stream.
       *  @param  __tiestr  The output stream.
       *  @return  The previously tied output stream, or NULL if the stream
       *           was not tied.
       *
       *  This sets up a new tie; see tie() for more.
      */
      basic_ostream<_CharT, _Traits>*
      tie(basic_ostream<_CharT, _Traits>* __tiestr)
      {
        basic_ostream<_CharT, _Traits>* __old = _M_tie;
        _M_tie = __tiestr;
        return __old;
      }

The only thing that cout.tie(NULL) does is that it sets a protected variable _M_tie to NULL (the business with __old should get optimized away by the compiler). We can check by calling cout.tie() without arguments that _M_tie already is NULL.

I can't check with winlibs (due to not having a Windows PC), but with normal GCC 13.2 at least, the only difference in the generated assembly is that cout.tie(NULL) generates a single instruction

  mov QWORD PTR std::cout[rip+224], 0

and we know that there already is a 0 at that address. So the statement really should have no effect.

My bet is that it's still some random fluctuation. Any Accepted submission with running time greater than (time limit) — 50 ms should be treated as "you got lucky". Codeforces actually reruns submissions that slightly exceed the time limit and takes the minimum.