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The US has distanced itself from an incursion into Russia - which Moscow says ended in the defeat of armed insurgents who entered from Ukraine, reports BBC.
Parts of the border region of Belgorod came under attack on Monday, in one of the largest cross-border raids since Russia invaded its neighbour last year.
Russia later released pictures of abandoned or damaged Western military vehicles, including US-made Humvees.
The US insisted it did not "encourage or enable strikes inside of Russia".
In a news briefing on Tuesday, Matthew Miller added: "It is up to Ukraine to decide how to conduct this war."
Some Ukrainian military experts and bloggers have suggested that the images of destroyed US vehicles released by Russia could have been staged.
Villages in Belgorod near the border were evacuated after coming under fire. Russia says 70 attackers were killed, and has insisted the fighters were Ukrainian.
Kyiv denies involvement - and two Russian paramilitary groups opposed to Russian President Vladimir Putin say they were behind the incursion.
Bd-pratidin English/Golam Rosul
