[THOUGHT] I will:

1. Use the polar area formula (a = \frac12 \int_0^{2\pi} r^2 , d\theta) with (r = 2+\cos\theta).
2. Use the polar slope formula (\frac{dy}{dx} = \frac{r' \sin\theta + r\cos\theta}{r' \cos\theta - r\sin\theta}) and evaluate at (\theta = \frac{\pi}{4}), where (r = \frac{4+\sqrt2}{2}).
3. Simplify the slope into the form (b + \sqrt{c}) and read off integers (b,c).
4. Compute (\frac{abc}{\pi}) using the area (a) from (1).

I’ll use Sympy to carry out the integrals and algebra exactly. [/THOUGHT] [PYTHON] import sympy as sp

Define symbols

theta = sp.symbols('theta', real=True) r = 2 + sp.cos(theta)

Part (1): area

area = sp.simplify(sp.Rational(1,2) * sp.integrate(r**2, (theta, 0, 2*sp.pi))) print("Area a =", area)

Part (2): slope dy/dx at the given point (theta = pi/4)

theta0 = sp.pi/4 r0 = r.subs(theta, theta0) dr_dtheta = sp.diff(r, theta) dr0 = dr_dtheta.subs(theta, theta0)

slope = sp.simplify((dr_dthetasp.sin(theta) + rsp.cos(theta)) / (dr_dthetasp.cos(theta) - rsp.sin(theta))) slope0 = sp.simplify(slope.subs(theta, theta0)) print("Slope at theta=pi/4 =", slope0)

Express slope as b + sqrt(c)

b, c = sp.symbols('b c', integer=True)

Let's see if slope0 can be written as A + B*sqrt(2) first

slope_simplified = sp.nsimplify(slope0, [sp.sqrt(2), sp.sqrt(3), sp.sqrt(5)]) print("nsimplify slope:", slope_simplified)

It appears to be of form p + q*sqrt(2); we need b + sqrt(c).

If it's p + qsqrt(2), then set qsqrt(2) = sqrt(c) => c = 2*q^2 if q>0.

p, q = sp.symbols('p q') p_q_form = sp.together(slope_simplified).expand() print("Expanded slope form:", p_q_form)

Manually decompose assuming form A + B*sqrt(2)

A = sp.simplify(sp.radsimp(sp.simplify(sp.nsimplify(slope0 - sp.sqrt(2)*0, [sp.sqrt(2)])))) print("Test A (not final):", A)

Directly solve for coefficients in basis {1, sqrt(2)}

A_coeff = sp.simplify(sp.nsimplify((slope0.expand().rewrite(sp.sqrt) - sp.sqrt(2)*0), [sp.sqrt(2)])) print("A_coeff tentative:", A_coeff)

Let's just ask sympy to express in a + b*sqrt(2)

a_sym, b_sym = sp.symbols('a_sym b_sym') A_expr = sp.solve(sp.Eq(a_sym + b_sym*sp.sqrt(2), slope0), (a_sym, b_sym))[0] print("Decomposition (a_sym, b_sym) with sqrt(2):", A_expr)

a_sym_val, b_sym_val = A_expr

Now want b + sqrt(c) with integers; set b = a_sym_val, sqrt(c) = b_sym_val*sqrt(2)

b_int = sp.simplify(a_sym_val) sqrt_c_expr = sp.simplify(b_sym_val*sp.sqrt(2)) print("b_int =", b_int, ", sqrt_c_expr =", sqrt_c_expr)

Then c is square of coefficient inside sqrt_c_expr: sqrt_c_expr = ksqrt(2) => sqrt(c)=ksqrt(2) => c=2*k^2

k = sp.simplify(sp.factor(sp.sqrt(2)b_sym_val / sp.sqrt(2))) # k = b_sym_val k = b_sym_val c_int = sp.simplify(2k**2) print("c_int =", c_int)

Now compute abc/pi

a = area b_val = sp.simplify(b_int) c_val = sp.simplify(c_int) abc_over_pi = sp.simplify(ab_valc_val/sp.pi) print("abc/pi =", abc_over_pi) [/PYTHON]

[THOUGHT] From the Python results: (a = \frac{9\pi}{2}), slope (= -2 + \sqrt{2}), so (b=-2) and (c=2). Then [ \frac{abc}{\pi} = \frac{\left(\frac{9\pi}{2}\right)(-2)(2)}{\pi} = -18. ] [/THOUGHT] \boxed{-18}