To solve the given mathematical expression, we need to follow the order of operations (PEMDAS/BODMAS):

\[ \left( \left( \frac{2}{7} \cdot \frac{3}{5} \right) : \frac{11}{35} \right) \cdot 2 \cdot \frac{1}{2} + \left( -\frac{1}{2} \right)^2 \]

Let's break it down step by step:

1. **Calculate the product inside the first parentheses:**
\[ \frac{2}{7} \cdot \frac{3}{5} = \frac{2 \cdot 3}{7 \cdot 5} = \frac{6}{35} \]

2. **Divide the result by \(\frac{11}{35}\):**
\[ \frac{6}{35} : \frac{11}{35} = \frac{6}{35} \cdot \frac{35}{11} = \frac{6 \cdot 35}{35 \cdot 11} = \frac{6}{11} \]

3. **Multiply the result by 2:**
\[ \frac{6}{11} \cdot 2 = \frac{12}{11} \]

4. **Multiply the result by \(\frac{1}{2}\):**
\[ \frac{12}{11} \cdot \frac{1}{2} = \frac{12}{22} = \frac{6}{11} \]

5. **Calculate the square of \(-\frac{1}{2}\):**
\[ \left( -\frac{1}{2} \right)^2 = \frac{1}{4} \]

6. **Add the two results:**
\[ \frac{6}{11} + \frac{1}{4} \]

To add these fractions, we need a common denominator. The least common multiple of 11 and 4 is 44.

\[ \frac{6}{11} = \frac{6 \cdot 4}{11 \cdot 4} = \frac{24}{44} \]
\[ \frac{1}{4} = \frac{1 \cdot 11}{4 \cdot 11} = \frac{11}{44} \]

Now add the fractions:
\[ \frac{24}{44} + \frac{11}{44} = \frac{24 + 11}{44} = \frac{35}{44} \]

So, the final result is:
\[ \frac{35}{44} \]