
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 $f(x)=2\cos \left( \frac {\pi}{2}-x\right)\sin x+(\sin x+\cos x)^{2}$.  
$(1)$ Find the intervals of monotonic increase for the function $f(x)$.  
$(2)$ Stretch all the x-coordinates of the points on the graph of $y=f(x)$ to twice their original values (the y-coordinates remain unchanged), then shift the resulting graph to the left by $\frac {\pi}{3}$ units to get the graph of the function $y=g(x)$. Calculate the value of $g\left( \frac {\pi}{6}\right)$.

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