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"In the first of what will be many tests for your new administration, Russia and its proxy forces launched attacks against Ukrainian forces this week, killing at least seven Ukrainian soldiers and wounding dozens more," McCain wrote in a letter to Trump released Thursday. "That this surge of attacks began the day after he talked with you by phone is a clear indication that Vladimir Putin is moving quickly to test you as commander-in-chief. America's response will have lasting consequences," The Hill reported.

Trump has expressed hope at improving relations with Russia and on Saturday spoke with Russian President Putin for the first time since Inauguration Day in what was described by the White House as a "positive call" that "was a significant start to improving the relationship."

The call included discussion of "the main aspects of the Ukrainian crisis," according to the Kremlin statement about the call, though the White House statement did not mention Ukraine.

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Read alsoAvdiyivka on fire: BM-21 GRAD MLRS position identified – volunteers (photo, video)On Sunday, violence flared in the Avdiyivka-Yasynuvata area and was described by some as the worst escalation since the conflict in eastern Ukraine began in 2014. Russia has denied backing the separatists, but western officials have said otherwise.

Under the annual defense policy bill, the U.S. president already has the authority to send so-called lethal aid to Ukraine such as weapons and ammunition. Former President Obama opted not to use the authority, instead sending "non-lethal" aid, such as economic assistance.

But McCain, a staunch Russia hawk, urged Trump to use the authority given to him by Congress.

"Vladimir Putin's violent campaign to destabilize and dismember the sovereign nation of Ukraine will not stop unless and until he meets a strong and determined response," wrote McCain, chairman of the Senate Armed Services Committee.

Read alsoReuters: U.S. makes limited exceptions to sanctions on Russian spy agencyMcCain also reiterated his call for Trump to maintain sanctions on Russia and impose new ones for Russia's interference in the U.S. election.

Failing to provide more help to Ukraine, McCain said, risks the country's sovereignty and American credibility.

"Ukrainians are not asking Americans to do their fighting," McCain said. "Nearly 10,000 Ukrainians have died to protect their homeland and many more are serving and have sacrificed for the cause of a free and united Ukraine. But America does have a proud history of helping free people to defend themselves."