SAN FRANCISCO — Outspoken billionaire Carl Icahn tossed another $500 million of Apple Inc. stock into his investment portfolio Thursday and suggested some new products as he tries to persuade the iPhone maker to do more to lift its market value.

With the latest shopping spree, Icahn has spent $1 billion on Apple stock during the past weeks to raise his total holdings in the Cupertino, Calif., company to $3.6 billion since he began buying the shares five months ago when they were still trading below $450. Apple’s stock rose $4.67 to close at $556.18 on Thursday. Despite the recent gains, the shares remain about 20 percent below their peak reached 16 months ago.

Icahn took the unusual step of pointing out new markets where he believes the company could easily make even more money than it already does with its line of iPhones, iPads, iPods and Mac computers.

Icahn also made it clear he wants to see an “ultra” high-definition Apple television set, a device that has been a subject of company speculation for the past three years.

He also touted the possibility of Apple making a smartwatch and thinks the company is ideally positioned to roll out a system for handling digital payments.

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