[THOUGHT] Find the parameter value t where the parametric curve intersects the line y=3x-5e, then compute the tangent slope dy/dx=(dy/dt)/(dx/dt) at that t. Finally match the slope to (aπ-b)/(π+c) and compute a+b+c. [/THOUGHT] [PYTHON] import sympy as sp

t = sp.symbols('t', real=True) x = sp.exp(4t) * (1 + sp.sin(sp.pit)**2) y = sp.exp(4t) * (1 - 3sp.cos(sp.pi*t)**2)

Solve intersection with y = 3x - 5e

eq = sp.Eq(y, 3x - 5sp.E) sol_t = sp.solve(eq, t) print("t solutions:", sol_t)

Choose the (real) solution and compute slope dy/dx there

dxdt = sp.diff(x, t) dydt = sp.diff(y, t)

for ts in sol_t: slope = sp.simplify((dydt/dxdt).subs(t, ts)) print("t =", ts, "slope =", slope)

Extract a,b,c from slope (3*pi-2)/(pi+6)

slope_simplified = sp.simplify((dydt/dxdt).subs(t, sp.Rational(1,4))) print("slope at t=1/4:", slope_simplified)

compute a+b+c assuming slope = (a*pi - b)/(pi + c)

a, b, c = 3, 2, 6 print("a+b+c =", a+b+c) [/PYTHON]

[THOUGHT] From the Python output, the slope is (\frac{3\pi-2}{\pi+6}), so (a=3), (b=2), (c=6) and (a+b+c=11). [ \boxed{11} ]