
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 $f(x)=\vec{a}\cdot \vec{b}$, where $\vec{a}=(\sqrt{2}\sin{\omega x},\sqrt{2}\cos{\omega x}+1)$ and $\vec{b}=(\sqrt{2}\cos{\omega x},\sqrt{2}\cos{\omega x}-1)$.
1. Find the range of the function $f(x)$ when $\omega=1$ and $x\in(0,\frac{\pi}{2})$.
2. Find the interval where the function $f(x)$ is monotonically decreasing when $\omega=-1$.

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