1. **State the problem:** Evaluate the expression $2\sec\left(\frac{3\pi}{4}\right) + \cot\left(\frac{\pi}{3}\right)$.\n\n2. **Recall the definitions and values:**\n- $\sec(\theta) = \frac{1}{\cos(\theta)}$.\n- $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.\n- Important angles: $\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$, $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$, $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.\n\n3. **Calculate $\sec\left(\frac{3\pi}{4}\right)$:**\n$$\sec\left(\frac{3\pi}{4}\right) = \frac{1}{\cos\left(\frac{3\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}.$$\nSimplify by rationalizing denominator:\n$$-\frac{2}{\sqrt{2}} = -\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}.$$\n\n4. **Calculate $\cot\left(\frac{\pi}{3}\right)$:**\n$$\cot\left(\frac{\pi}{3}\right) = \frac{\cos\left(\frac{\pi}{3}\right)}{\sin\left(\frac{\pi}{3}\right)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}}.$$\nRationalize denominator:\n$$\frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}.$$\n\n5. **Combine the results:**\n$$2\sec\left(\frac{3\pi}{4}\right) + \cot\left(\frac{\pi}{3}\right) = 2(-\sqrt{2}) + \frac{\sqrt{3}}{3} = -2\sqrt{2} + \frac{\sqrt{3}}{3}.$$\n\n**Final answer:**\n$$\boxed{-2\sqrt{2} + \frac{\sqrt{3}}{3}}.$$