
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 $\tan(2\alpha) = \frac{3}{4}$, with $\alpha \in \left(- \frac{\pi}{2}, \frac{\pi}{2}\right)$, let $f(x) = \sin(x + \alpha) + \sin(\alpha - x) - 2\sin\alpha$, and for any $x \in \mathbb{R}$, it always holds that $f(x) \geq 0$. Find the value of $\sin\left(\alpha - \frac{\pi}{4}\right)$.

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