
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 functions $f(x)=\sqrt{3}\sin (\pi x+\frac{\pi }{3})$ and $g(x)=\sin (\frac{\pi }{6}-\pi x)$, let $M$ and $N$ denote the points where the graphs of the functions intersect the $y$-axis on its left and right sides respectively. If $O$ represents the origin, find $\overrightarrow{OM}\cdot \overrightarrow{ON}$.

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