import math

PRECISION = 24 * 60

def average_time(times: list[str]) -> str:
    sum_sin = 0.0
    sum_cos = 0.0
    for time_str in times:
        hours, mins = map(int, time_str.split(':'))
        total_mins = (hours % 24) * 60 + mins
        angle = 2 * math.pi * total_mins / PRECISION
        sum_sin += math.sin(angle)
        sum_cos += math.cos(angle)
    # Compute average angle
    avg_angle = math.atan2(sum_sin, sum_cos)
    # Convert back to minutes
    avg_mins = round((avg_angle * PRECISION) / (2 * math.pi))
    # Adjust for negative values
    avg_mins = avg_mins % PRECISION
    # Convert to hours and minutes
    hours = int(avg_mins // 60)
    minutes = int(avg_mins % 60)
    return f"{hours:02d}:{minutes:02d}"

# Example usage:
print(average_time(["23:45", "00:17"]))