## Step 1: Determine the current time and the current day of the week Since the problem doesn't specify the current day, we need to identify the day based on the context. Given that the future planning session is described to follow after the next meeting and lunch break, it's logical to infer that we must be counting backwards from today, but the problem doesn't specify what day yet. ## Step 2: Determine the conflict and timing lead-up to fixing the future meeting The conflict occurred at 2:30 PM, and since we're trying to delay it in the future, we need to count backwards, assuming "tomorrow" means the next day. ## Step 3: Calculate the time of the meeting that needs to be rescheduled Since the meeting was rescheduled from 2:30 PM to 4:15 PM yesterday, and if we were in the 24-hour format, 4:15 PM + 30 minutes = 5:45 PM is the scheduled time. Backwards it would mean we count down to 3:45 PM, so it is 3:45 PM yesterday. ## Step 4: Determine the schedule for the future planning session After counting backwards from today's time considering lunchtime: 12:30 PM was lunch, so we count backwards to find the past meeting time: lunchtime then the 9 AM meeting, then back an hour to 8 AM for the planning session. ## Step 5: Determine the start time of the planning session With the meeting rescheduled for after lunch, after counting backwards, the start of the future planning session is the 7 AM tomorrow. ## Step 6: Convert the scheduling to 24-hour format Converting 7 AM to 7:00 to meet the format requirement. ## Step 7: Account for the lunch break's start time Accounting for the lunch break with the calculated 7:00 AM start of the planning session but skipping lunch until the 12:30 PM break. ## Step 8: Finalize with the correct afternoon appointment The final arrangement is 7:00 AM - 12:30 PM. ## Step 9: Provide the time in 24-hour format as required. Combine the result of step 8 with the earlier adjusted lunch break for exact 24-hour clock. The lunch break is from 12:30 PM to 1:30 PM and so does not change the night's (AM) time. The start is given as 7:00 AM as accurate to the problem statement. The final answer is: $\boxed{07:00}$