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As soon as the engine starts to die, get out and hold the coil wire 1/4" from a head nut to check for a hot blue spark. If it's good, then look at the fuel supply.
It sounds like it runs low on fuel to me. Remove the fuel line at the carb, see how far the lone extends beyond the ferrule. It should be no more than about 1/8", or it may hit the filter screen and block off some flow. Let the fuel flow into a quart jar to see that it can maintain a good strong flow for several minutes.
If the flow checks out, then take the carb apart to check the filter screen, jets, and passages. If you have a known good carb to swap, that would be a quick easy way to know also.