Triangle $ABC$ has integer side lengths. Rectangles $BCDE, ACFG, ABHJ$ are constructed so that $CD = AC + AB$, $CF = AB + BC$, and $BH = (AC + BC)^2$. If $[ABHJ] = [BCDE] + [ACFG]$, how many different values can $[ABC]$ have?

**Details and assumptions**

$[PQRS]$ refers to the area of figure $PQRS$.

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