
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 $f(x)= \sqrt{3}\sin (x+ \frac{\pi}{3})-\cos x$.

$(1)$ Find the minimum value of $f(x)$ in the interval $[0,\pi]$;

$(2)$ Given $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, with $b=5 \sqrt{3}$ and $\cos A= \frac{3}{5}$. If $f(B)=1$, find the length of side $a$.

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