It　is　well　known　that　the　classic　Allen-Cahn　equation　satisfies　the　maximum　bound　principle　(MBP),　that　is,　the　absolute　value　of　its　solution　is　uniformly　bounded　for　all　time　by　certain　constant　under　suitable　initial　and　boundary　conditions.　In　this　paper,　we　consider　numerical　solutions　of　the　modified　Allen-Cahn　equation　with　a　Lagrange　multiplier　of　nonlocal　and　local　effects,　which　not　only　shares　the　same　MBP　as　the　original　Allen-Cahn　equation　but　also　conserves　the　mass　exactly.　We　reformulate　the　model　equation　with　a　linear　stabilizing　technique,　then　construct　first-　and　second-order　exponential　time　differencing　schemes　for　its　time　integration.　We　prove　the　unconditional　MBP　preservation　and　mass　conservation　of　the　proposed　schemes　in　the　time　discrete　sense　and　derive　their　error　estimates　under　some　regularity　assumptions.　Various　numerical　experiments　in　two　and　three　dimensions　are　also　conducted　to　verify　the　theoretical　results.