
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
In the graph of function $f\left(x\right)= \dfrac{1}{2}\sin \left(2\omega x+\varphi \right)$ ($\omega  \gt 0$, $|\varphi | \lt  \dfrac{\pi }{2}$), which is shifted to the right by $\dfrac{\pi }{12}$ units to obtain the graph of $g\left(x\right)$, the graph of $g\left(x\right)$ is symmetric about the origin. For vector $\overrightarrow {m}=\left( \sqrt {3}\sin \omega x,\cos 2\omega x\right)$ and $\overrightarrow {n}=\left( \dfrac{1}{2}\cos \omega x, \dfrac{1}{4}\right)$, where $\omega  \gt 0$, $f\left(x\right)= \overrightarrow {m}\cdot  \overrightarrow {n}$. Choose one of the following three conditions: ① function $f\left(x\right)=\cos \omega x\sin \left(\omega x+ \dfrac{\pi }{6}\right)- \dfrac{1}{4}$, and supplement it in the question below, then solve.

Given ___, the distance between two adjacent axes of symmetry of the graph of function $f\left(x\right)$ is $\dfrac{\pi }{2}$.

(1) If $0 \lt \theta  \lt  \dfrac{\pi }{2}$, and $\sin \theta = \dfrac{\sqrt {2}}{2}$, find the value of $f\left(\theta \right)$.

(2) Find the intervals where the function $f\left(x\right)$ is monotonically decreasing on the interval $\left[0,2\pi \right]$.

Note: If multiple conditions are chosen and answered separately, the first answer will be scored.

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