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Prior studies on intelligent reflecting surface (IRS) have mostly considered wireless communication systems aided by a single passive IRS, which, however, has limited control over wireless propagation environment and suffers severe product-distance path-loss. To address these issues, we propose in this paper a new multi-active multi-passive (MAMP)-IRS aided wireless communication system, where a number of active and passive IRSs are deployed to assist the downlink communication in complex environment, by establishing a multi-hop reflection path across active and passive IRSs. An optimization problem is formulated to maximize the achievable rate of a typical user by designing the active-and-passive IRS routing path as well as the joint beamforming of the BS and selected active/passive IRSs. To draw useful insights into the optimal design, we first consider a special case of the single-active multi-passive (SAMP)-IRS aided system. For this case, we propose an efficient algorithm to obtain its optimal solution by first optimizing the joint beamforming given any SAMP-IRS routing path, and then optimizing the routing path by using a new path decomposition method and graph theory. Next, for the general MAMP-IRS aided system, we show that its challenging beam routing optimization problem can be efficiently solved by a new two-phase approach. Its key idea is to first optimize the inner passive-IRS beam routing between each two active IRSs for effective channel power gain maximization, followed by an outer active-IRS beam routing optimization for rate maximization. Last, numerical results are provided to demonstrate the effectiveness of the proposed MAMP-IRS beam routing scheme.