
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 that the maximum value of $f(x)=\sin (2014x+ \frac {π}{6})+ \cos (2014x- \frac {π}{3})$ is $A$, find the minimum value of $A|x_{1}-x_{2}|$, where $x_{1}$, $x_{2}$ are real numbers such that for any real number $x$, $f(x_{1}) \leqslant f(x) \leqslant f(x_{2})$ always holds.

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