
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that $ \cos\left(x- \frac{\pi}{4}\right) = \frac{\sqrt{2}}{10}$, where $x\in\left(\frac{\pi}{2}, \frac{3\pi}{4}\right)$:
$(1)$ Find the value of $\sin x$.
$(2)$ Find the value of $\sin\left(2x + \frac{\pi}{3}\right)$.

Present the answer in LaTex format: \boxed{Your answer}
user