
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 $f\left( x \right)=2\cos \left( \omega x+\dfrac{\pi }{6} \right)\left( \omega > 0,x\in R \right)$ with the smallest positive period of $10\pi$.

(1) Find the equation of the axis of symmetry for the function $f(x)$;

(2) Let $\alpha ,\beta \in \left[ 0,\dfrac{\pi }{2} \right],$ and given $f\left( 5\alpha +\dfrac{5\pi }{3} \right)=-\dfrac{6}{5},f\left( 5\beta - \dfrac{5\pi }{6} \right)=\dfrac{16}{17},$ find the value of $\cos \left( \alpha +\beta \right)$.

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