The value of $3\frac{2}{3}÷2\frac{1}{5}\times \frac{1}{4}\times \left(1+\frac{3}{1-\frac{2}{5}}\right)$ is:

Convert the mixed fractions into improper fractions:
* $3\frac{2}{3}=\frac{11}{3}$
* $2\frac{1}{5}=\frac{11}{5}$

Now, simplify the expression step-by-step following the BODMAS rule:
First, evaluate the term inside the parentheses:

$$\left(1+\frac{3}{1-\frac{2}{5}}\right)=1+\frac{3}{\frac{3}{5}}=1+3\times \frac{5}{3}=1+5=6$$

Substitute this back into the main expression:

$$\frac{11}{3}÷\frac{11}{5}\times \frac{1}{4}\times 6$$

Perform the division:

$$\frac{11}{3}\times \frac{5}{11}\times \frac{1}{4}\times 6$$

$$=\frac{5}{3}\times \frac{1}{4}\times 6$$

$$=\frac{30}{12}=\frac{5}{2}=2\frac{1}{2}$$