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We can help each other while competing with each other.

How should I avoid the redundancy of 'each other' here?
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I don't think you can and still maintain clarity of expression.
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Absolutely impossible, unless you mean you and I will help each other and compete against these OTHER people.
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OK, so we need the 'each other' there again. Good.
And it's impossible to replace it with 'them', right? 
 BarbaraPA's reply was promoted to an answer.
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Good! Thanks, GG!