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Inter's first decent sight of goal came shortly before the half-hour mark when Hernanes created space for himself to shoot from outside the box, firing an effort just past the post.
Hernanes was able to use that effort as a sighter in the 37th minute when he latched on to a knockdown in the area, beating one defender inside the box before slamming the ball past the keeper.
Palacio almost doubled his personal tally moments after the restart when he latched onto a pass inside the box, but this time the keeper was equal to his effort.
Inter came close three times in quick succession with a little over 10 minutes of the match remaining as first Mauro Icardi and then Fredy Guarin saw efforts blocked, while the latter also turned a rebound wide.
The hosts almost levelled things up in the 82nd minute when Marco Benassi did well to create space for himself to shoot from range, but his effort flew narrowly over the bar.
They wouldn't have to wait much longer to restore parity, however, finding the equaliser with just five minutes remaining courtesy of a fine solo effort from Innocent Emeghara.
Neither side could find a winner in the closing stages as Inter missed the chance to close the gap on fourth, while Livorno moved one point closer to safety.