Project: s41467-025-56140-1 | Function: accel_profile (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Generates the velocity profile for 3D printer motion by calculating acceleration, steady-state, and deceleration phases for each movement segment. This implements the core T-Code algorithm for time-based synchronization by creating the motion profile needed to map auxiliary commands to precise timestamps.

**KEY TERMS:** {
  "accel_dist_dict": "acceleration phase distances",
  "steady_state_dist": "constant velocity phase distance",
  "decel_dist": "deceleration phase distance",
  "accel_time_abs_dict": "absolute time profile for motion",
  "flag_accel_result": "optimization cache for acceleration calculations"
}

**CONTEXT CODE:**                 Z_var_string_start = Z_var[0] + str(Z_start)
                for elem in Z_var[1:]:
                    Z_var_string = Z_var_string + ', '+ elem
                    Z_var_string_0 = Z_var_string_0 + ' ' + elem + str(0)
                    Z_var_string_start = Z_var_string_start + ' ' + elem + str(Z_start)

                line = line.replace('{Z_var_list}', Z_var_string).replace('{ramprate}', str(accel)).replace('{feed}',str(feed))
                line = line.replace('{Z_var_0}', Z_var_string_0)
                line = line.replace('{Z_var_Z_start}', Z_var_string_start)


                f.write(line)
            g.close()

        f.write('\nDwell(3)\n\r')

        for i in range(len(Sum_coord_dict)):
            G_command = Sum_G_command_dict[i]
            # if G_command == "G3":
            #     G_command = "CCW"
            # if G_command == "G2":
            #     G_command = "CW"
            coordinates = str(G_command) + " "
            for j in range(len(Sum_coord_dict[i])):
                dist = Sum_coord_dict[i][j]
                variable = Sum_var_dict[i][j]
                if variable == 'Z':
                    variable = Z_var[0]
                if variable != 0:
                    if variable == Z_var[0]:
                        for elem in Z_var:
                            coordinates += str(elem) + str(dist) + " "
                    else:
                        coordinates += str(variable) + str(dist) + " "


            f.write("\n\r" + coordinates)

            if i != 0 and i < len(Sum_coord_dict) - 1 and i in resync_trigger_distance_dict:
            # if i in resync_trigger_distance_dict:
                f.write(resync_PING)

        f.write('\n\rDwell(5)')
        f.write('\n\rHome(['+Z_var_string+ '])')

        f.close()

    return f

## creates acceleration profile
def accel_profile(Sum_distance_dict):
    # MASKED: Generates the velocity profile for 3D printer motion by calculating acceleration, steady-state, and deceleration phases for each movement segment. This implements the core T-Code algorithm for time-based synchronization by creating the motion profile needed to map auxiliary commands to precise timestamps.
    raise NotImplementedError("This function is masked for testing purposes.")

## creates dictionary of time and commands
def distance2time(accel_profile_distance, accel_profile_time, feed, accel, decel, distance_commands_dict, locate_resync):
    time_list = []
    time_dict = {}
    x_current = 0
    t = 0
    distance = 0
    index_start = 0
    flag_rel_dist_accel = {}
    flag_rel_dist_max_vel = {}
    flag_rel_dist_decel = {}
    flag_count = 0
    count = 0
    time_resync = 0
    command_list = []
    times_of_resync = []
    for j in range(len(distance_commands_dict)):
        if type(distance_commands_dict[j]) == str:
            #time_dict[j] = distance_commands_dict[j]
            command_list.append(distance_commands_dict[j])
            time_dict[t] = command_list
        else:
            command_list = []
            distance += abs(distance_commands_dict[j])
            for i in range(index_start, len(accel_profile_distance)):
                accel_region = accel_profile_distance[i][0]
                max_velocity_region = accel_profile_distance[i][1]
                decel_region = accel_profile_distance[i][2]

                if distance >= decel_region:
                    t_current = accel_profile_time[i][2] - time_resync
                    x_current = decel_region

                    t = t_current
                    index_start += 1

                elif distance >= max_velocity_region:
                    t_current = accel_profile_time[i][1] - time_resync
                    x_current = max_velocity_region
                    t = t_current

                elif distance >= accel_region:
                    t_current = accel_profile_time[i][0] - time_resync
                    x_current = accel_region
                    t = t_current

                if distance < accel_region:
                    relative_distance = distance - x_current
                    try:
                        relative_distance = round(relative_distance, 7)
                        t_current = flag_rel_dist_accel[relative_distance]
                    except:
                        t_current = findt(0, accel, relative_distance)
                        relative_distance = round(relative_distance, 7)

                    t += t_current
                    flag_rel_dist_accel[relative_distance] = t_current
                    break

                elif distance < max_velocity_region:
                    count += 1
                    relative_distance = distance - x_current
                    try:
                        relative_distance = round(relative_distance, 7)
                        t_current = flag_rel_dist_max_vel[relative_distance]
                        flag_count += 1
                    except:
                        t_current = relative_distance / feed  # findt(feed, 0, relative_distance)
                        relative_distance = round(relative_distance, 7)
                    t += t_current
                    flag_rel_dist_max_vel[relative_distance] = t_current
                    break

                elif distance < decel_region:
                    relative_distance = distance - x_current

                    try:
                        relative_distance = round(relative_distance, 7)
                        t_current = flag_rel_dist_decel[relative_distance]
                    except:
                        t_current = findt(feed, decel, relative_distance)
                        relative_distance = round(relative_distance, 7)

                    t += t_current
                    flag_rel_dist_decel[relative_distance] = t_current
                    break

                if distance == decel_region or distance == max_velocity_region or distance == accel_region:
                    break

            time_output = t
            time_list.append(time_output)
            if j in locate_resync:
                times_of_resync.append(t)

            #time_dict[j] = time_output


Complete the function implementation replacing "raise NotImplementedError".