Participate & Involve, are fairly interchangeable, whilst engage tends to relate to conversation, or action of a joint nature. It really all depends on the surrounding context, for example, in a dispute between management and employees: * Management agreed to participate in talks with employees; * Management agreed to involve themselves in talks with employees; * Management agreed to engage in talks with employees; Engage would be the most suitable. Or, if the employees are arranging a fun day: * Management agreed to participate in the preparations for the employee fun day * Management agreed to be involved in the preparations for the employee fun day * Management agreed to engage with the employees in preparation for the employee fun day.
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