To solve the expression −57−(−567)+1330+12+(−156)+1314−(−25)-\frac{5}{7} - \left(-\frac{5}{67}\right) + \frac{13}{30} + \frac{1}{2} + \left(-1 \frac{5}{6}\right) + 1 \frac{3}{14} - \left(-\frac{2}{5}\right)−75​−(−675​)+3013​+21​+(−165​)+1143​−(−52​), we will simplify each term step by step.

−57−(−567)-\frac{5}{7} - \left(-\frac{5}{67}\right)−75​−(−675​):

−57+567=−5⋅677⋅67+5⋅767⋅7=−335469+35469=−300469-\frac{5}{7} + \frac{5}{67} = -\frac{5 \cdot 67}{7 \cdot 67} + \frac{5 \cdot 7}{67 \cdot 7} = -\frac{335}{469} + \frac{35}{469} = -\frac{300}{469}−75​+675​=−7⋅675⋅67​+67⋅75⋅7​=−469335​+46935​=−469300​
1330\frac{13}{30}3013​ (already in simplest form).
12\frac{1}{2}21​ (already in simplest form).
−156-1 \frac{5}{6}−165​:

−156=−(1+56)=−66−56=−116-1 \frac{5}{6} = -\left(1 + \frac{5}{6}\right) = -\frac{6}{6} - \frac{5}{6} = -\frac{11}{6}−165​=−(1+65​)=−66​−65​=−611​
13141 \frac{3}{14}1143​:

1314=1+314=1414+314=17141 \frac{3}{14} = 1 + \frac{3}{14} = \frac{14}{14} + \frac{3}{14} = \frac{17}{14}1143​=1+143​=1414​+143​=1417​
−(−25)-\left(-\frac{2}{5}\right)−(−52​):

−(−25)=25-\left(-\frac{2}{5}\right) = \frac{2}{5}−(−52​)=52​
Now, combine all the simplified terms together:

−300469+1330+12−116+1714+25-\frac{300}{469} + \frac{13}{30} + \frac{1}{2} - \frac{11}{6} + \frac{17}{14} + \frac{2}{5}−469300​+3013​+21​−611​+1417​+52​To add these fractions, find a common denominator, which is 210:

−300469=−300⋅30469⋅30=−900014070-\frac{300}{469} = -\frac{300 \cdot 30}{469 \cdot 30} = -\frac{9000}{14070}−469300​=−469⋅30300⋅30​=−140709000​ 1330=13⋅730⋅7=91210\frac{13}{30} = \frac{13 \cdot 7}{30 \cdot 7} = \frac{91}{210}3013​=30⋅713⋅7​=21091​ 12=105210\frac{1}{2} = \frac{105}{210}21​=210105​ −116=−35210-\frac{11}{6} = -\frac{35}{210}−611​=−21035​ 1714=255210\frac{17}{14} = \frac{255}{210}1417​=210255​ 25=42210\frac{2}{5} = \frac{42}{210}52​=21042​Now add all these fractions together:

−900014070+91210+105210−35210+255210+42210-\frac{9000}{14070} + \frac{91}{210} + \frac{105}{210} - \frac{35}{210} + \frac{255}{210} + \frac{42}{210}−140709000​+21091​+210105​−21035​+210255​+21042​Combine the numerators:

−900014070+91+105−35+255+42210=−900014070+458210-\frac{9000}{14070} + \frac{91 + 105 - 35 + 255 + 42}{210} = -\frac{9000}{14070} + \frac{458}{210}−140709000​+21091+105−35+255+42​=−140709000​+210458​Convert 458210\frac{458}{210}210458​ to a fraction with a common denominator:

458210=458⋅67210⋅67=3068614070\frac{458}{210} = \frac{458 \cdot 67}{210 \cdot 67} = \frac{30686}{14070}210458​=210⋅67458⋅67​=1407030686​Now subtract the fractions:

−900014070+3068614070=2168614070-\frac{9000}{14070} + \frac{30686}{14070} = \frac{21686}{14070}−140709000​+1407030686​=1407021686​Therefore, the simplified result of the expression is 2168614070\boxed{\frac{21686}{14070}}1407021686​​.