
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 $f(x)=\cos^2{x}-\sin^2{x}+ \frac{1}{2}$, for $x \in (0,\pi)$:
$(1)$ Find the intervals where $f(x)$ is monotonically increasing;
$(2)$ Let $\triangle ABC$ be an acute-angled triangle, with the side opposite angle $A$ being $a= \sqrt {19}$ and the side opposite angle $B$ being $b=5$. If $f(A)=0$, find the area of $\triangle ABC$.

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