Put 12/1/2013 in F2. Format for mmmm yyyy if you like. These formulas work regardless of formatting. Then this formula in F3 and fill down.
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=DATE(YEAR($F$2),MONTH($F$2)+ROWS($1:1),DAY($F$2))
Then this formula in G2 and fill down.
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=COUNTIFS($B$2:$B$1403,">="&$F2,$C$2:$C$1403,"<="&EOMONTH($F2,0))