Interpret TwoMega

In this challenge, you will write an interpreter for 2^Ω (transcribed as TwoMega), a language based loosely on brainfuck with an infinite-dimensional storage space.

The Language

2^Ω contains three pieces of state:

• The Tape, which is an infinite list of bits, all initialized to 0. It has a leftmost element, but no rightmost element.

• The Memory Pointer, which is a nonnegative integer that is an index of an element in the tape. A higher memory pointer refers to a tape cell further to the right; a memory pointer of 0 refers to the leftmost element. The memory pointer is initialized to 0.

• The Hypercube, which is a conceptually ∞-dimensional "box" of cells, each of which contains a bit initialized to 0. The width of the Hypercube is bound in every dimension to only 2 cells, but the infinity of dimensions means the number of cells is uncountable.

An index into the hypercube is an infinite list of bits that refers to a cell in the hypercube (in the same way that a finite list of bits could be used to refer to a hypercube of finite dimension). Because the tape is an infinite list of bits, the entire tape always refers to an element of the Hypercube; this element is called the referent.

2^Ω gives meaning to 7 different characters:

• < decrements the memory pointer by 1. Decrementing it below 0 is undefined behavior, so you do not need to handle it.
• > increments the memory pointer by 1.
• ! flips the bit at the referent.
• . outputs the bit at the referent.
• ^ replaces the bit at the cell pointed to by the memory pointer on the tape with the inverse of the bit at the referent.
• [x] runs the code x as long as the bit at the referent is 1.

The Challenge

Your task is to write a program that takes a string as input and executes that input as a 2^Ω program.

This is code-golf, so the shortest valid answer (measured in bytes) wins.

Notes

• You can assume that the program will consist solely of the characters <>!.^[] and that [] will be properly nested.
• Your interpreter should only be limited by available memory on the system. It should be able to run the sample programs in a reasonable amount of time.

Sample Programs

Print 1:

!.

Print 010:

.!.!.

Print 0 forever:

![!.!]

Print 0 forever, or 1 forever if ! is prepended:

[.]![!.!]