
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 $f(x)=\sin x \cdot \cos x - \sqrt{3}\cos^2 x$,
1. Find the smallest positive period of $f(x)$ and its intervals of monotonic increase;
2. If the graph of function $f(x)$ is transformed such that every point's x-coordinate is stretched to twice its original value while the y-coordinate remains unchanged, resulting in the graph of function $g(x)$, and if the equation $g(x)+ \frac{\sqrt{3}+m}{2}=0$ has solutions for $x \in [0,\pi]$, find the range of the real number $m$.

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