
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 the function $f(x) = \sin\left(\frac {5\pi}{6} - 2x\right) - 2\sin(x - \frac{\pi}{4})\cos(x + \frac{3\pi}{4})$.
1. Find the least positive period of $f(x)$ and its intervals of monotonic increase.
2. If $x \in \left[\frac{\pi}{12}, \frac{\pi}{3}\right]$ and the minimum value of $F(x) = -4\lambda f(x) - \cos(4x - \frac{\pi}{3})$ is $-\frac{3}{2}$, find the value of the real number $\lambda$.

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