from __future__ import annotations

from typing import Tuple

def admm_step(x: float, y: float, u: float, rho: float, a: float, b: float) -> Tuple[float, float, float]:
    # Simple quadratic objective example:
    # minimize 0.5*a*x^2 + 0.5*b*y^2
    # with constraint x + y = 1
    # Standard ADMM update on a toy problem to illustrate API:
    # x^{k+1} = argmin_x L_gamma(x, y^k, u^k)
    # For demonstration, perform a plain projection step toward constraint and dual update.
    # This is intentionally tiny and not a production solver.
    # Update primal via a simple gradient-descent-like projection
    x_new = (1.0 - y)  # enforce x+y=1
    y_new = 1.0 - x_new

    # Dual ascent step (additional stability term)
    u_new = u + rho * (x_new + y_new - 1.0)
    return float(x_new), float(y_new), float(u_new)