1. **State the problem:** Simplify the expression $$ (2^{1}x^{0} y^{2})^{-3} \cdot 2 y x^{3} $$ and then simplify $$ (x^{-3})^{4} x^{4} $$.\n\n2. **Recall the rules:**\n- Any number or variable raised to the zero power is 1, so $x^{0} = 1$.\n- Power of a power: $(a^{m})^{n} = a^{m \cdot n}$.\n- Negative exponent: $a^{-m} = \frac{1}{a^{m}}$.\n- When multiplying like bases, add exponents: $a^{m} \cdot a^{n} = a^{m+n}$.\n\n3. **Simplify the first expression:**\nStart with $ (2^{1} x^{0} y^{2})^{-3} \cdot 2 y x^{3} $.\nSince $x^{0} = 1$, this becomes $ (2 \cdot y^{2})^{-3} \cdot 2 y x^{3} $.\nApply the negative exponent: $ (2)^{-3} (y^{2})^{-3} = 2^{-3} y^{-6} $.\nSo the expression is $ 2^{-3} y^{-6} \cdot 2 y x^{3} $.\nMultiply the coefficients: $2^{-3} \cdot 2 = 2^{-3+1} = 2^{-2}$.\nMultiply the $y$ terms: $y^{-6} \cdot y^{1} = y^{-5}$.\nThe $x$ term is $x^{3}$.\nSo the simplified expression is $$ 2^{-2} y^{-5} x^{3} = \frac{x^{3}}{2^{2} y^{5}} = \frac{x^{3}}{4 y^{5}} $$.\n\n4. **Simplify the second expression:**\nStart with $ (x^{-3})^{4} x^{4} $.\nApply power of a power: $x^{-3 \cdot 4} = x^{-12}$.\nMultiply by $x^{4}$: $x^{-12} \cdot x^{4} = x^{-12+4} = x^{-8}$.\nRewrite with positive exponent: $$ x^{-8} = \frac{1}{x^{8}} $$.\n\n**Final answers:**\n- First expression: $$ \frac{x^{3}}{4 y^{5}} $$.\n- Second expression: $$ \frac{1}{x^{8}} $$