
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 $f\left(x\right)=\cos x\left(\sin x+\cos x\right)-\frac{1}{2}$.  
$(1)$ If $0 \lt \alpha\ \ \lt \frac{π}{2}$ and $\sin \alpha =\frac{\sqrt{2}}{2}$, find the value of $f\left(\alpha \right)$.  
$(2)$ Find the intervals where the function $f\left(x\right)$ is monotonically increasing.

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