
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 $f(x)=\sqrt{3}\sin^{2}x+\sin x \cdot \cos x$.  
$(1)$ Find the interval where $f(x)$ is monotonically increasing.  
$(2)$ If for any $x \in [t,\frac{\pi}{3}]$ it holds that $|f(x)-\frac{\sqrt{3}}{2}| \leq \frac{\sqrt{3}}{2}$, find the range of real number $t$.

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