A sign at the entrance to the Edward Hines Jr. VA Hospital outside Chicago. (Scott Olson/Getty Images)

The policy-making body of the American Medical Association is meeting this week in Chicago, where it is debating a number of important resolutions. Here's one you won't find on the agenda.

"Be it resolved that every licensed physician in the United States should volunteer to offer one appointment, free of charge, to a U.S. veteran who has been waiting at least 90 days to see a doctor at a Veterans Administration facility."

Is that as outrageous as it sounds? There were 834,769 licensed physicians in this country in 2012, according to this table from the Henry J. Kaiser Foundation, many with access to some of the finest medical facilities in the world. At the same time, we learned Monday, some 57,000 of the men and women who who fought or supported wars in Afghanistan, Iraq, Vietnam, Korea and elsewhere have waited at least 90 days to see a VA doctor. And another 64,000 have sought care but haven't even been able to get on VA waiting lists.

If even one-seventh of U.S. doctors agreed to see a vet once for half an hour or an hour, the system could triage that backlog, possibly in fewer than the 14 days the VA considers a reasonable time to wait for an appointment. (Even less if retired physicians or nurse practitioners or physicians' assistants were brought into the mix).

At VA facilities, an overburdened, inefficient system is trying to cope with a big influx of veterans. It will take months to untangle this mess, (even as the agency is beginning to adopt the same idea of sending some patients to private physicians). Let the private docs do it. They can see vets at their practices. They can see them in hospitals. They can see them in clinics. They can offer an inital assessment, hear what's wrong, get the ball rolling.

Yes, there is a doctor shortage in the United States, especially in primary care. Yet this idea is not unprecedented. Some doctors see poor patients pro bono or at very reduced rates. And health fairs at which physicians and dentists see the homeless or uninsured free have long been a staple in communities across the country.

I know, I know. There are a million reasons why this could never happen. Why should doctors offer their services free of charge to fix a huge government screw-up? Matching vets and doctors would be an administrative nightmare. Vets would have to sign waivers assuring physicians that they aren't liable for anything that goes wrong. How would patients get the follow-up care that many undoubtedly need, not to mention lab tests and prescription medications? How would doctors communicate what they found to the VA medical staff?

But honestly, could those logistics be any more difficult than the shame of sending men and women to war and then failing to provide them medical care? What if one life-threatening condition was diagnosed? Or critically needed hypertension medication prescribed? Or a diabetic's wound treated before he lost a toe? Or dialysis averted? Or a vet with PTSD got to speak with a psychiatrist for an hour?

One vet, one doctor, one visit. Unless someone has a better idea.