For the people that are talking about the weekly and nightfall this is not an issue there since those are not match made games. And if you have to leave and come back on another character for the regular strikes then something is wrong with you because they are so easy and simple. The timer will not work either because some people will come and follow through the zones but not help you.
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a timer/warp into battle feature would work because most will likely just fight at that point, not all but if forced to actually use there controller to avoid dying/sit there to move every time it warps them forward, most will just play out the strike since it'll go by faster at that point if they are required to sit at there system
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I have found quite a few people who have put more effort into hiding and doing nothing than it would take to just do the strike. This is why I have stopped doing strikes all together. And also because they are to repetitive and rewards are not that great wish they was a way to make them better. Like maybe add different strikes with different modifiers on.