Inequality involving a kind of Harmonic mean

While revising the Harmonic mean, I came across this inequality which I haven't figured out how to solve, but I think it should be the application of some known inequality. I would be very grateful if anyone could give me a suggestion how to solve it.

Consider a tuple of $$n$$ positive real numbers $$(x_1,x_2,\dots,x_n)$$ and another tuple of $$n$$ positive real numbers $$(y_1,y_2,\dots,y_n)$$. Let us denote as $$\bar{x}$$ the mean of the first tuple, i.e. $$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$$, and with $$\bar{y}$$ the mean of the second tuple $$\bar{y} = \frac{\sum_{i=1}^n y_i}{n}$$.

Is it possible to find the largest positive real number $$C$$ such that

$$$$\frac{1}{n} \sum_{i=1}^n {\frac{1}{\frac{1}{x_i}+\frac{1}{y_i}}} \geq C \cdot {\frac{1}{\frac{1}{\bar{x}}+\frac{1}{\bar{y}}}}$$$$

is always verified ?

For odd $$i$$ let $$x_i\rightarrow0^+$$ and $$y_i=1$$;
For even $$i$$ let $$x_i=1$$ and $$y_i\rightarrow0^+$$.
Thus, $$C\leq0$$ and the largest positive $$C$$, for which your inequality is true does not exist.
By the way, the reversed inequality is true for $$C=1$$.