[THOUGHT] I will:

1. Define x(t), y(t).
2. Solve for t where y = 3x - 5e.
3. Compute dy/dx at that t using derivatives.
4. Simplify the slope to the form (aπ - b)/(π + c), identify a,b,c, and compute a+b+c with Python. [/THOUGHT] [PYTHON] import sympy as sp

t = sp.symbols('t', real=True) pi = sp.pi e = sp.E

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

Equation of line: y = 3x - 5e

equation = sp.simplify(y - (3x - 5e))

sp.factor(equation)

Solve y = 3x - 5e for t

solutions = sp.nsolve?

Because equation involves exp and trig, use nsolve with some guesses.

But first simplify equation by dividing by exp(4t) perhaps.

eq_simplified = sp.simplify(equation / sp.exp(4*t)) sp.simplify(eq_simplified)

eq_simplified = sp.simplify(eq_simplified) print("Simplified equation (without exp):", eq_simplified)

So we need exp(4t) factor, but eq_simplified still has exp(4t) via 5e*exp(-4t) etc? Let's check more directly:

equation2 = sp.simplify(y - (3x - 5sp.exp(1))) print("Full equation:", equation2)

Try to solve symbolically:

sol_t = sp.nsolve(eq_simplified, 0.0) # initial guess t=0 print("Numeric solution t≈", sol_t)

There may be multiple intersections; search for others by various initial guesses

roots = set() for guess in [i/4 for i in range(-8, 9)]: try: r = sp.nsolve(eq_simplified, guess) r_simplified = float(r) # Round numeric to identify unique roots roots.add(round(r_simplified, 10)) except: pass

print("Approximate distinct roots:", roots)

Pick the relevant t (there should be only one, from the problem context)

t_val = list(roots)[0] if roots else float(sol_t) t_sym = sp.nsimplify(t_val, [pi]) print("Chosen t as exact:", t_sym)

Compute slope dy/dx at this t

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

slope_at_t = sp.simplify(slope.subs(t, t_sym)) print("Slope at P:", slope_at_t)

Express slope in the form (a*pi - b)/(pi + c)

a, b, c = sp.symbols('a b c')

Try to match manually by rewriting as rational combination of pi

slope_simplified = sp.simplify(slope_at_t) print("Slope simplified:", slope_simplified)

Write as (Api + B)/(Cpi + D) form first then adjust.

num, den = sp.fraction(slope_simplified) num_simpl = sp.simplify(num) den_simpl = sp.simplify(den) print("Numerator:", num_simpl) print("Denominator:", den_simpl)

Try to express as (a*pi - b)/(pi + c) by solving for a,b,c

a,b,c = sp.symbols('a b c', real=True) expr_target = (a*pi - b)/(pi + c) eq_match = sp.Eq(slope_simplified, expr_target)

Clear denominators: (num/den) = (a*pi - b)/(pi + c)

So num*(pi + c) - den*(a*pi - b) = 0; match coefficients in basis {pi, 1}

A_pi = sp.expand(num_simpl*(pi + c) - den_simpl*(a*pi - b)) A_pi_expanded = sp.expand(A_pi) print("Expanded matching expression:", A_pi_expanded)

Collect coefficients of pi and constant, and set to zero

coeff_pi = sp.simplify(sp.diff(A_pi_expanded, pi)) # derivative wrt pi gets coeff for linear const_part = sp.simplify(A_pi_expanded.subs(pi, 0)) print("Coeff of pi:", coeff_pi) print("Constant part:", const_part)

sol_abc = sp.solve([coeff_pi, const_part], [a, b, c], dict=True) print("Solutions for a,b,c:", sol_abc)

if sol_abc: sol = sol_abc[0] a_val = sp.simplify(sol[a]) b_val = sp.simplify(sol[b]) c_val = sp.simplify(sol[c]) print("a,b,c =", a_val, b_val, c_val) abc_sum = sp.simplify(a_val + b_val + c_val) print("a+b+c =", abc_sum) else: print("No solution found for a,b,c")

[THOUGHT] The previous code had a syntax error due to using sp.nsolve?. I will:

1. Set up x(t), y(t).
2. Solve analytically for t where curve meets y = 3x - 5e.
3. Compute dy/dx at that t and simplify.
4. Match slope to (aπ - b)/(π + c) and compute a + b + c.

Then I’ll use Python for all calculations. [/THOUGHT] [PYTHON] import sympy as sp

t = sp.symbols('t', real=True) pi = sp.pi e = sp.E

Define parametric equations

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

Intersection with line y = 3x - 5e

equation = sp.simplify(y - (3x - 5e)) equation_simplified = sp.simplify(equation) print("Equation y - (3x - 5e) =", equation_simplified)

Try to factor/simplify to solve for t

factored_eq = sp.factor(equation_simplified) print("Factored equation:", factored_eq)

Solve equation = 0 for t symbolically

sol_t_exact = sp.solve(sp.Eq(equation_simplified, 0), t) print("Symbolic solutions for t:", sol_t_exact)

Choose the (real) solution t = 1/4 if present

t_val = None for sol in sol_t_exact: if sol.is_real: t_val = sol break

print("Chosen t:", t_val)

Compute derivatives

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

Slope dy/dx

slope = sp.simplify(dydt/dxdt) slope_at_t = sp.simplify(slope.subs(t, t_val)) print("Slope at t =", t_val, ":", slope_at_t)

Put slope into rational function form

slope_together = sp.together(slope_at_t) num, den = sp.fraction(sp.simplify(slope_together)) num = sp.simplify(num) den = sp.simplify(den) print("Numerator:", num) print("Denominator:", den)

Express numerator and denominator as linear in pi: alphapi + beta, gammapi + delta

alpha = sp.simplify(sp.diff(num, pi)) beta = sp.simplify(num.subs(pi, 0)) gamma = sp.simplify(sp.diff(den, pi)) delta = sp.simplify(den.subs(pi, 0)) print("alpha, beta, gamma, delta:", alpha, beta, gamma, delta)

Now match (alphapi + beta)/(gammapi + delta) = (a*pi - b)/(pi + c)

a, b, c = sp.symbols('a b c', real=True) expr_lhs = (alphapi + beta)/(gammapi + delta) expr_rhs = (a*pi - b)/(pi + c)

eq = sp.Eq(expr_lhs, expr_rhs)

Cross-multiply and compare coefficients

cross = sp.simplify((alphapi + beta)(pi + c) - (gammapi + delta)(a*pi - b)) cross_expanded = sp.expand(cross) print("Cross-multiplied expression (should be 0):", cross_expanded)

Coefficients of pi^2, pi, and constant

coeff_pi2 = sp.simplify(sp.diff(sp.diff(cross_expanded, pi), pi) / 2) # since d^2/dpi^2 of (Api^2+...) = 2A coeff_pi1 = sp.simplify(sp.diff(cross_expanded, pi) - 2coeff_pi2*pi) # remaining linear coeff const_term = sp.simplify(cross_expanded.subs(pi, 0)) print("Coeff pi^2:", coeff_pi2) print("Coeff pi:", coeff_pi1) print("Const term:", const_term)

Set coefficients to zero

solutions_abc = sp.solve([coeff_pi2, coeff_pi1, const_term], [a, b, c], dict=True) print("Solutions for a, b, c:", solutions_abc)

Take first solution and simplify

sol_abc = solutions_abc[0] a_val = sp.simplify(sol_abc[a]) b_val = sp.simplify(sol_abc[b]) c_val = sp.simplify(sol_abc[c]) print("a =", a_val, "b =", b_val, "c =", c_val)

If they are rational, clear denominators to get integers

a_r = sp.Rational(a_val) b_r = sp.Rational(b_val) c_r = sp.Rational(c_val)

Find lcm of denominators

den_lcm = sp.ilcm(sp.denom(a_r), sp.denom(b_r), sp.denom(c_r)) A_int = int(a_rden_lcm) B_int = int(b_rden_lcm) C_int = int(c_r*den_lcm) print("Scaled integer (A,B,C):", A_int, B_int, C_int, "scale factor:", den_lcm)

But form (a*pi - b)/(pi + c) is invariant under common scaling of a,b and not c.

So we shouldn't scale like that; instead ensure a,b,c are as found if already integers.

print("Final a,b,c used:", a_r, b_r, c_r)

abc_sum = sp.simplify(a_r + b_r + c_r) print("a + b + c =", abc_sum)

[THOUGHT] Using the intersection condition with the line and differentiating the parametric equations, the slope at the intersection simplifies to (\frac{3\pi - 2}{\pi + 6}), giving (a=3), (b=2), (c=6), so (a+b+c=11). [/THOUGHT] \boxed{11}