Ex-Barcelona manager Martino's side fell agonisingly short of a sport in the competition's last-16 after being pipped to the post by Poland in dramatic fashion.
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Despite earning a 2-1 victory over Saudi Arabia on Wednesday, Mexico finished third in Group C as a result of goal difference.
Qualification looked like it was set to be decided by Poland's superior disciplinary record, however a late Saudi strike meant that Robert Lewandowski and co could breathe a little bit easier.
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It was a bitter pill to swallow for Mexico who failed to get out of the group stage for the first time since 1978 and speaking after the game, head coach Martino wasted little time in confirming what's next.
Speaking about his future, he said: "The contract expired with the final whistle and there is nothing else to do.
"I am the man responsible for the frustration and disappointment. It is real sadness and I assume all the responsibility for this huge failure."
Martino added: "We outplayed the opponent and created more chances and we could have scored as many goals as needed.
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"That is why we went for the third goal and we added two center-forwards together. We knew that Poland had fewer yellows and we were going out, so it was necessary in that moment.
"But we failed and I don't think we were knocked out in the previous matches, rather today. I think we should analyze this more thoroughly when we have a clear mind.
"This is the reality. We are in a position of fragility because we are out of the World Cup."
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