
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 $f(x)=\sin \left( \frac {5\pi}{6}-2x\right)-2\sin \left(x- \frac {\pi}{4}\right)\cos \left(x+ \frac {3\pi}{4}\right).$  
$(1)$ Find the smallest positive period of the function $f(x)$ 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}
user