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Once it's in the news, it's too late for that, sort of like when deep pockets show up at the poker game, the limits get too high.

Push back from the table and ease on out.
As you pretty well know, Old TexMex, here there are two ways that the regular, everyday Joe has available to deal with "problems".
1. The first way -- and always preferable -- is to "trade favors". What can I do for you to get you to do what I want? Maybe I can do nothing, but perhaps I have a friend who can. Often the promise of a simple good word or introduction is enough. Other times, you need to up the ante. This option only works if you have
something to trade as a favor that you are willing to trade. But it is almost always the best option when dealing with Mexicans who have something you want.
2. Throw money at it. This will always work, but the cost is often repugnant and keeps going up everytime you are required to repeat the action to solve a problem. In fact, once the Mexicans -- whose National Sport is removing money from the foreigners -- find out that
you are willing to throw money at a problem, they'll keep making sure you keep having problems so you will keep throwing money. It's a vicious circle. I am always amazed at how many regular, everyday people from the U.S. and Canada just never seem to be able to figure this out because they always want the solution applied NOW, with no delay. The Mexicans of course know that, and use the impatience to their advantage to force a money solution over a favor-trade. But the favor-trade is the way it's almost always done by people who live here day-to-day.
I do not like to sound callous when talking about the life and well-being of a fellow human being, but Old TexMex is completely right when he says it has hit the News and now pride has become involved on both sides. I hope that the "officials" dealing with the Mexican side have an actual clue about dealing with Mexicans -- but I fear they will not.
This is a high-stakes poker game now, and probably was from the instant he was detained. The first ante at the table should probably have been something like; "Well, who -- or what -- do we have that you want?" Then let the bargaining begin.
An unwillingness to "play the game" will only lead to throwing money at the problem. You have no idea how often over the last near-quarter-century of living here I have had this discussion with Canadians, Americans, or Europeans and even some South Americans and Asians who were in a bit of a sticky spot due to some deal gone slightly bad with the Mexicans. I have found that even close Canadian friends who have no reason to doubt my word often disbelieve me simply because they are completely unwilling to look at the glass as a Mexican looks at it. That always leads to the money-throw, and it's never the Mexicans doing the throwing.
I hope this turns out positively for the detained Marine but I fear it will become a pride-and-media circus with him just left sitting there to suffer while talking heads who are at the end of the day clueless about doing actual on-the-ground business with Mexicans chat up the airwaves with everything they are doing on his behalf. It's sad. And yes, I am angry that this sort of thing happens and I really do see it all the time as the years go by. If you're going to deal with the Mexicans and can't explain to some disinterested third party how exactly they are looking at the glass, then don't go into the deal because they are not looking at it the same way you are.
If you can explain how they are looking at it (despite whatever claims they might make to having the same vision as you) well then go ahead, and proceed with caution and don't put anything on the table you can't afford to lose. As I said; sad.