
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 $f(x)=\sin x\cos (x+\frac{π}{3})$.  
$(1)$ Find the interval where $f(x)$ is monotonically increasing.  
$(2)$ In triangle $ABC$, with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, and angle $A$ being acute. If $f(A)=-\frac{{\sqrt{3}}}{4}$, $a=\sqrt{5}$, and $b+c=3$, find the area of triangle $ABC$.

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