Architect a generative system that builds complex self similar fractal s pc 1539

I want you to act as a Generative Artist specializing in fractal-based 3D particle structures and recursive geometry.

Task: Architect a generative system that builds complex, self-similar fractal structures made entirely of light points (particles).

Design Specifications:

Use a recursive algorithm (like a Mandelbulb or Sierpinski gasket) to define the initial coordinates of the particle cloud.

Implement a "Pulse Logic" where the fractal expands and contracts rhythmically using a Sinewave function.

Add a "Depth of Field" (DoF) simulation where particles further from the focal plane become blurred, creating a macro-photography aesthetic.

Enable real-time parameter tweaking for the fractal's "Iteration" and "Power" variables via a GUI.

Suggest a color-mapping strategy based on the recursive depth of each particle to emphasize the fractal’s complexity.

Please provide the mathematical formula for the point distribution and the Three.js setup for the PointsMaterial and Depth effect.