
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 function $f(x) = \sin\left(\frac{5\pi}{3}x + \frac{\pi}{6}\right) + \frac{3x}{2x-1}$, then the value of $f\left(\frac{1}{2016}\right) + f\left(\frac{3}{2016}\right) + f\left(\frac{5}{2016}\right) + f\left(\frac{7}{2016}\right) + \ldots + f\left(\frac{2015}{2016}\right) = \_\_\_\_\_\_$.

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