To solve the given expression, we need to follow the order of operations, which includes simplifying inside the parentheses first and then performing the division.

The given expression is:
\[ \left( \frac{4}{5} \cdot \frac{12}{13} \cdot \frac{10}{36} \right) \div \left( -\frac{6}{13} \right) \]

First, simplify the multiplication inside the parentheses:
\[ \frac{4}{5} \cdot \frac{12}{13} \cdot \frac{10}{36} \]

Multiply the numerators and the denominators:
\[ \frac{4 \cdot 12 \cdot 10}{5 \cdot 13 \cdot 36} \]

Calculate the products:
\[ \frac{480}{2340} \]

Simplify the fraction by finding the greatest common divisor (GCD) of 480 and 2340, which is 60:
\[ \frac{480 \div 60}{2340 \div 60} = \frac{8}{39} \]

Now, divide this result by \(-\frac{6}{13}\):
\[ \frac{8}{39} \div -\frac{6}{13} \]

Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{8}{39} \cdot -\frac{13}{6} \]

Multiply the numerators and the denominators:
\[ \frac{8 \cdot -13}{39 \cdot 6} = \frac{-104}{234} \]

Simplify the fraction by finding the greatest common divisor (GCD) of 104 and 234, which is 26:
\[ \frac{-104 \div 26}{234 \div 26} = \frac{-4}{9} \]

So, the final result is:
\[ \boxed{-\frac{4}{9}} \]