[THOUGHT] We need to find the point P where the parametric curve intersects the line y = 3x - 5e. That means we need to find t such that y(t) = 3x(t) - 5e. Then compute the slope dy/dx at that t. The slope is dy/dx = (dy/dt)/(dx/dt). Then express slope as (aπ - b)/(π + c) and compute a+b+c.

Steps:

1. Define x(t) = exp(4t)(1 + sin^2(πt)), y(t) = exp(4t)(1 - 3 cos^2(πt)).

2. Solve y(t) = 3x(t) - 5e for t. That is: exp(4t)(1 - 3 cos^2(πt)) = 3exp(4t)(1 + sin^2(πt)) - 5e. Simplify: exp(4t)(1 - 3 cos^2(πt)) = 3 exp(4t)(1 + sin^2(πt)) - 5e. Rearr: 3 exp(4t)(1 + sin^2(πt)) - exp(4t)(1 - 3 cos^2(πt)) = 5e. => exp(4t)[3(1 + sin^2(πt)) - (1 - 3 cos^2(πt))] = 5e. Compute inside: 3 + 3 sin^2(πt) - 1 + 3 cos^2(πt) = 2 + 3(sin^2(πt) + cos^2(πt)) = 2 + 31 = 5. So exp(4t)5 = 5e => exp(4t) = e => 4t = 1 => t = 1/4. Indeed, at t = 1/4, we have x = e^{4(1/4)}(1 + sin^2(π/4)) = e(1 + (√2/2)^2) = e*(1 + 1/2) = (3/2)e. y = e^{1}(1 - 3 cos^2(π/4)) = e(1 - 3*(1/2)) = e*(1 - 3/2) = e*(-1/2) = -e/2. Check line: 3x - 5e = 3*(3/2)e - 5e = (9/2 - 5)e = (9/2 - 10/2)e = (-1/2)e = y. Good.

3. Compute dx/dt and dy/dt at t = 1/4. x(t) = e^{4t}(1 + sin^2(πt)). Let f(t) = e^{4t}, g(t) = 1 + sin^2(πt). Then x = fg. dx/dt = f'g + fg'. f' = 4 e^{4t}. g' = 2 sin(πt) cos(πt) * π = π sin(2πt) (since 2 sin θ cos θ = sin 2θ). So dx/dt = 4 e^{4t}*(1 + sin^2(πt)) + e^{4t} * π sin(2πt).

   y(t) = e^{4t}(1 - 3 cos^2(πt)). Let h(t) = 1 - 3 cos^2(πt). h' = -3 * 2 cos(πt) * (-sin(πt)) * π? Actually derivative of cos^2(πt) = 2 cos(πt) * (-π sin(πt)) = -2π cos(πt) sin(πt). Multiply by -3: h' = -3 * (-2π cos(πt) sin(πt)) = 6π cos(πt) sin(πt) = 3π sin(2πt). dy/dt = 4 e^{4t}(1 - 3 cos^2(πt)) + e^{4t} * 3π sin(2πt).

   At t = 1/4: sin(π/4) = √2/2, cos(π/4) = √2/2. sin^2 = 1/2, cos^2 = 1/2. sin(2πt) = sin(π/2) = 1. e^{4t} = e.

   Compute: x = e*(1 + 1/2) = (3/2)e. y = e*(1 - 3*(1/2)) = e*(1 - 3/2) = -e/2.

   dx/dt = 4e*(1 + 1/2) + e * π * 1 = 4e*(3/2) + eπ = 6e + eπ = e(6 + π). dy/dt = 4e*(1 - 3*(1/2)) + e * 3π * 1 = 4e*(1 - 3/2) + 3eπ = 4e*(-1/2) + 3eπ = -2e + 3eπ = e(3π - 2).

   Slope m = dy/dx = (dy/dt)/(dx/dt) = e(3π - 2) / (e(6 + π)) = (3π - 2)/(π + 6).

   Compare with given form (aπ - b)/(π + c). Here numerator: 3π - 2 => a=3, b=2. Denominator: π + 6 => c=6. So a+b+c = 3+2+6 = 11.

But we should verify with Python to avoid algebra mistakes.

[/THOUGHT] [PYTHON] import sympy as sp

Define symbols

t = sp.symbols('t')

Define functions

x = sp.exp(4t) * (1 + sp.sin(sp.pi * t)**2) y = sp.exp(4t) * (1 - 3*sp.cos(sp.pi * t)**2)

Solve intersection with line y = 3x - 5*e

e = sp.E equation = sp.Eq(y, 3x - 5e) solution_t = sp.solve(equation, t) print("t solutions:", solution_t)

Pick t = 1/4 (should be only real solution)

t_val = sp.Rational(1,4) print("t =", t_val)

Compute derivatives

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

Evaluate at t = 1/4

dx_val = dxdt.subs(t, t_val).simplify() dy_val = dydt.subs(t, t_val).simplify() print("dx/dt at t = 1/4:", dx_val) print("dy/dt at t = 1/4:", dy_val)

Slope

m = dy_val / dx_val print("Slope m =", m)

Simplify to form (aπ - b)/(π + c)

m_simplified = sp.simplify(m) print("Simplified m =", m_simplified)

Extract a, b, c

Write m as (a*pi - b)/(pi + c)

We can match numerator and denominator

num, den = m_simplified.as_numer_denom() print("Numerator:", num) print("Denominator:", den)

Expect numerator = a*pi - b, denominator = pi + c

Solve for a, b, c by matching coefficients of pi and constant

Let's write numerator as Api + B, denominator as Cpi + D

But denominator is pi + c, so C=1, D=c.

Actually denominator is pi + 6, so c=6.

Numerator is 3*pi - 2, so a=3, b=2.

Compute a+b+c

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

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