Round corners (fillet) in Illustrator CS6

I have the following figure in which I need to make a fillet to each its corners:

Is there some tool to avoid doing this manually? I tried the Effect > Stylize > Round Corners tool but it apply the roundness also to the circles.
I'm using the CS6 version of Adobe Illustrator.

Answer

There's no easy way to do this nativley in Illustrator CS6. (In the last couple of CC editions Adobe implemented a tool called Live corners which does it)

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If you just used the Rounded Corner effect, it warps the circle (see below)

You have to do a bit of prep first:

Add in a line (dashed in my case) in a locked background layer (to align both my anchor points evenly)

Then place 2 anchor points there (one on each side), and only then can you use the Effects → Rounded Corners

The problem with first doing rounded corners effect, is that it causes the circle to get warped (because there aren't any anchor points, if you add them in, it stops the rounding at that anchor point)

_{Notice, that I did not add in anchor points to the bottom part, and it got warped I've added a new gif}

Update: in my image it seems as though it is warping the path. here are the outcomes (after rounding corner effect):

Overlayed at 30% (after that anchor point, none of the rest of the path is effected):

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For a much easier and simpler method, I would recommend a great plug-in by Astute Graphics* called Vectorscribe.

It is very simple and easy to use. (It isn't free, however they offer a 14 day free trial)

It also has some other tools included in vectorscribe, like measurements and path cleanup

_{Vectorscribe in action:}

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_{*I'm not affiliated with Astute Graphics}