Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$.

[i]Proposed by Andrew Wu[/i]
1. **Identify the given elements and construct the points \( P \) and \( Q \):**
   - Given triangle \( ABC \) with \( AB = 6\sqrt{3} \), \( BC = 14 \), and \( CA = 22 \).
   - Incenter \( I \) and circumcircle \( \Gamma \).
   - Points \( P \) and \( Q \) are constructed on rays \( BA \) and \( CA \) such that \( BP = CQ = 14 \).

2. **Analyze the angles and properties of the triangle:**
   - Since \( BP = CQ = 14 \), we need to find the relationship between the angles and sides.
   - Note that \( \angle BPX = \angle BPI = \angle CBI = \frac{1}{2} \angle C \) because \( I \) is the incenter.
   - Similarly, \( \angle BXP = 180^\circ - \angle BPX - \angle PBX = 180^\circ - \frac{1}{2} \angle C - \angle PBX \).

3. **Determine the lengths \( BX \) and \( CY \):**
   - Since \( \triangle BPX \) is isosceles with \( BP = BX = 14 \), we have \( BX = 14 \).
   - Similarly, \( CY = 14 \).

4. **Use the properties of the tangents and similar triangles:**
   - The tangents from \( B \) and \( C \) to \( \Gamma \) intersect at point \( R \).
   - Since \( BX = BP = 14 \) and \( CY = CQ = 14 \), we can use the similarity of triangles.
   - \( \triangle RXY \sim \triangle RBC \).

5. **Calculate the length \( BR \) and \( CR \):**
   - Since \( BR = CR \) and the tangents from a point to a circle are equal, we have \( BR = CR = \frac{14}{\sqrt{3}} \).

6. **Find the length \( XY \) using similar triangles:**
   - Using the similarity \( \triangle RXY \sim \triangle RBC \), we have:
     \[
     \frac{XY}{BC} = \frac{RX}{RB} = \frac{RY}{RC}
     \]
   - Since \( BC = 14 \), we get:
     \[
     XY = 14 \sqrt{3} - 14
     \]

7. **Sum the values of \( a \), \( b \), and \( c \):**
   - Given \( XY = a\sqrt{b} - c \), we identify \( a = 14 \), \( b = 3 \), and \( c = 14 \).
   - Therefore, \( a + b + c = 14 + 3 + 14 = 31 \).

The final answer is \( \boxed{31} \).