Find the value of the following expression:
$$3\frac{2}{3}÷2\frac{1}{5}\times \frac{1}{4}\times \left(1+\frac{3}{1-\frac{2}{5}}\right)$$

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

Now substitute these into the expression:

$$\frac{11}{3}÷\frac{11}{5}\times \frac{1}{4}\times \left(1+\frac{3}{1-\frac{2}{5}}\right)$$

Step 1: Solve the expression inside the parenthesis
First, simplify the denominator inside the parenthesis:

$$1-\frac{2}{5}=\frac{3}{5}$$

Now solve the complex fraction:

$$\frac{3}{\frac{3}{5}}=3\times \frac{5}{3}=5$$

Add $1$ to this result:

$$1+5=6$$

Step 2: Solve the main expression using BODMAS rules
Now the expression is:

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

Perform the division first:

$$\frac{11}{3}÷\frac{11}{5}=\frac{11}{3}\times \frac{5}{11}=\frac{5}{3}$$

Now perform the multiplications sequentially from left to right:

$$\frac{5}{3}\times \frac{1}{4}=\frac{5}{12}$$

$$\frac{5}{12}\times 6=\frac{5}{2}=2\frac{1}{2}$$