
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 vectors $\overrightarrow{a}=(\cos ωx,\sin ωx)$ and $\overrightarrow{b}=(\cos ωx, \sqrt {3}\cos ωx)$, where $ω > 0$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}- \dfrac {1}{2}$ with the smallest positive period of $π$.
1. Find the expression for the function $f(x)$ and its interval of decrease;
2. In $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively, and $S$ be its area. If $f(\dfrac{A}{2})=1$, $b=1$, and $S_{\triangle ABC}=\sqrt{3}$, find the value of $a$.

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