
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
The graph of the function $f(x)=\sin (\omega x+ \frac {\pi}{6})$ ($\omega > 0$) intersects the $x$-axis at points whose $x$-coordinates form an arithmetic sequence with a common difference of $\frac {\pi}{2}$. To obtain the graph of $g(x)=\cos (\omega x+ \frac {\pi}{6})$, determine the horizontal shift applied to the graph of $f(x)$.

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