Exploring a Triangle - Medians, Altitudes and Angle Bisectors!

Let $m_a, h_a, w_a$ denote the lengths of the median, the altitude and the internal angle bisector, respectively, to side $A$ in a $\Delta ABC$ . Define $m_b, m_c, h_b, h_c, w_b, w_c$ similarly. Let $R$ be the circumradius of $\Delta ABC$ . Let:

• $\eta_1 = \max \left( \displaystyle \dfrac{1}{R} \sum_{cyclic} \dfrac{b^2+c^2}{m_a} \right)$

• $\eta_2 = \min \left( \displaystyle \dfrac{1}{R} \sum_{cyclic} \dfrac{b^2+c^2}{h_a} \right)$

• $\eta_3 = \min \left( \displaystyle \dfrac{1}{R} \sum_{cyclic} \dfrac{b^2+c^2}{w_a} \right)$

Find the value of $\large{ \left \lfloor \eta_1 + \eta_2 + \eta_3 \right \rfloor}$ .