By now, President Obama's remark that "we don't have a strategy yet" has made the rounds. Republicans were quick to pounce on it, as well they should have.

But while the White House went into damage-control mode, emphasizing that it was a reference to the lack of decisions about increasing military action in Iraq and/or Syria and not a lack of a broader strategy there, the damage was already done.

As with all gaffes, the worst ones are the ones that confirm people's preexisting suspicions or fit into an easy narrative. That's why "47 percent" stung Mitt Romney so much, and it's why "don't have a strategy" hurts Obama today.

Polls have increasingly shown that Americans view Obama as a weak commander in chief without much direction or heft t0 his foreign policy. The latest is a Pew Research Center survey, released shortly before Obama's errant statement Thursday, that showed 54 percent of Americans say he's "not tough enough" when it comes to foreign policy and national security.

Just 36 percent said Obama has shown about the right amount of toughness. Mind you, this is after he launched airstrikes in Iraq.

As the graph above shows, Obama is regressing on this measure. And the same poll showed just 35 percent of Americans approve of Obama when it comes to both Iraq and the Ukraine/Russia situation – both below his overall approval rating of 42 percent.

Of course, people saying you’re not tough enough or disapproving of your job performance aren’t the same things as them saying you don’t have a strategy. But all of these things are intertwined, and there are plenty of data points to suggest people don’t think Obama has a particularly firm grasp on his strategy for dealing with world affairs.

A recent Washington Post-ABC News poll showed just 41 percent thought Obama was a “good manager,” while other polls from CNN and Pew both showed people increasingly don’t think he can get things done or “manage government effectively.”

Republicans, meanwhile, have criticized Obama for years as being reactive rather than proactive, often using the “leading from behind” verbiage that an anonymous White House official had the poor judgment to utter to the New Yorker's Ryan Lizza.

And as a series of overseas foreign policy crises have popped up in recent years, the White House has remained largely hands-off -- a decision that rightly or wrongly feeds the narrative of it not having a real strategy. And it certainly didn’t help that the White House set a so-called “red line” of the Syrian government using chemical weapons on its people, but didn’t actually do anything when it crossed the red line.

And then Obama goes and says something like he did Thursday.

Will this gaffe push a whole bunch of voters into the GOP column in November? Of course not. But it certainly helps the GOP make the case that Obama’s foreign policy continues to “lead from behind.” And to the extent foreign policy matters in the coming election (which it’s starting to look like it could), that could put some red-state Democrats in tough positions.

More so, though, this strikes us as a legacy problem for Obama. For a president confronting a bunch of overseas crises in the final two-plus years of his presidency -- including ones that involve or could involve U.S. force -- “we don’t have a strategy yet” could become pretty unhelpful shorthand for his foreign policy if things don’t go well.

Kind of like “lead from behind” -- or “47 percent.”