Bullshit. How often do those "critical deadlines" end up not meaning anything other than some MBA chose a date at random? And if the business truly does hinge on meeting this one deadline, why is he he only one on it?
As for your sob story, why in the holy fuck did you put off the work until the week of the meeting?
As for your sob story, why in the holy fuck did you put off the work until the week of the meeting?