Step 1: Understanding the coagulation process
Blood exists as a negatively charged sol. When ferric chloride (FeCl$_3$) is introduced, the Fe$^{3+}$ ions interact with the negatively charged blood sol, causing it to coagulate. This occurs due to the electrostatic attraction between the positively charged Fe$^{3+}$ ions and the negatively charged blood particles.
Step 2: The role of \(Fe^{3+} ions\)
The Fe$^{3+}$ ions neutralize the negative charge on the blood particles, leading to their aggregation and coagulation. This explains why Fe$^{3+}$ ions are responsible for facilitating the coagulation of blood.
Let \( \alpha = \dfrac{-1 + i\sqrt{3}}{2} \) and \( \beta = \dfrac{-1 - i\sqrt{3}}{2} \), where \( i = \sqrt{-1} \). If
\[ (7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}, \] then the value of \( m \) is ___________.