
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 $\overrightarrow{a}=(\cos x,-\frac{1}{2})$ and $\overrightarrow{b}=(\sin x+\cos x,1)$, with $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$:

1. If $0<\alpha<\frac{\pi}{2}$ and $\sin\alpha=\frac{\sqrt{2}}{2}$, find the value of $f(\alpha)$.
2. Find the smallest positive period and the monotonically increasing interval of the function $f(x)$.

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