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* Yen gains broadly on geopolitical worries
* Euro-yen sinks for 11th straight day, touches 4-month lows
* French presidential election also weighs (Updates to U.S. market open, changes dateline; previous LONDON)
By Dion Rabouin
NEW YORK, April 11 (Reuters) - The dollar fell broadly on Tuesday as geopolitical risk and declining U.S. Treasury yields pushed traders out of the greenback.
The Japanese yen rose broadly, as possible U.S. action in Syria and North Korea, and a resurgence of a previously written-off leftist contender in France's presidential race prompted buying of the safe-haven currency.
The possibility of some kind of U.S. military action against North Korea in response to its weapons tests grew after U.S. missile strikes against Syria last week in retaliation for a chemical weapons attack on civilians.
U.S. Treasury yields fell for a second straight day as anxiety increased.
Buying of Treasuries and the yen also jumped after Reuters reported black smoke was rising from part of an airfield in Moscow. The smoke turned out to be caused by gas burning nearby, and the move was a "knee-jerk reaction," analysts said.
The yen rose as much as 0.6 percent to 110.21 yen against the dollar in early North American trading. The dollar was last trading at 110.31 yen.
"It shows how nervous markets are about the geopolitical situation," said Marc Chandler, global head of currency strategy at Brown Brothers Harriman. "And at the end of the day the focus on the geopolitical data is coming without much serious economic data."
The euro also fell against the yen as investors weighed the possibility of a face-off between far-right candidate Marine Le Pen and left-wing candidate Jean-Luc Melenchon, who has surged in polls recently.
If the euro holds Tuesday's downturn, it would mark 11 straight days of losses against the Japanese currency, the longest streak in the history of the euro, Chandler noted.
"We see further downside as risks ahead of the first round of the French presidential elections have grown of late... so all that (is) boosting demand for safe havens," said Valentin Marinov, head of FX strategy at Credit Agricole in London.
Both Le Pen's and Melenchon's pledges to hold referendums on France's membership in the European Union have sparked fears among investors of the potential for a wider breakup of the euro zone.
The dollar index, which gauges the U.S. currency against a basket of six major peers, was off 0.25 percent on the day at 100.75.
"The dollar rally we've seen over the past week may be a bit overdone considering the risks we're still facing," said Thu Lan Nguyen, currency analyst with Commerzbank in Frankfurt. (Reporting by Dion Rabouin; Additional reporting by Ritvik Carvalho; Editing by Dan Grebler)