Simplify the terms inside the main square root.

Step 1: Simplify the terms inside the main square root.
The expression is $\sqrt{13 \sqrt{2^{2026}} + 2^{2027} + 4^{1013}}$.

First, simplify the innermost square root:

$$\sqrt{2^{2026}} = (2^{2026})^{\frac{1}{2}} = 2^{\frac{2026}{2}} = 2^{1013}$$

Next, express $4^{1013}$ with base 2:

$$$$