Task:
Evaluate $\int\int_C (x+y)^2 dx,dy$, where C is the circle defined by $x^2 + y^2 = M$, and M is the birth year of Li Bai.
Output format requirement: Round to three decimal places, using half up rounding rule.

Assistant:
<thought> To solve this problem, let's start by determining Li Bai's birth year. </thought>
<execute> print(wikipedia_search('Li Bai')) </execute>

Observation:
Page: Li Bai
Summary: Li Bai (Chinese: 李白; pinyin: Lǐ Bái, 701–762), also pronounced as Li Bo, courtesy name Taibai (Chinese: 太白), was a Chinese poet, acclaimed from his own time to the present as a brilliant and romantic figure who took traditional poetic forms to new heights.

Assistant:
<thought>
From the wiki_search result, we can see that Li Bai was born in 701, thus M = 701.
To evaluate the double integral of $(x+y)^2$ over the region $C$, where $C$ is the circle defined by $x^2 + y^2 = 701$, we should use polar coordinates.
In polar coordinates, $x = r\cos(\theta)$ and $y = r\sin(\theta)$, and the differential area element $dx,dy$ becomes $r,dr,d\theta$.
$\int\int_C (x+y)^2 dx,dy = \int_0^{2\pi}\int_0^{\sqrt{701}} (r\cos(\theta) + r\sin(\theta))^2r,dr,d\theta$
Let's use the SymPy package to calculate this.
</thought>
<execute>
from sympy import symbols, integrate, cos, sin, pi

r, theta = symbols('r theta')

# the integrand
integrand = (r*cos(theta) + r*sin(theta))**2 * r

# perform the double integration
double_integral = integrate(integrate(integrand, (r, 0, 701 ** 0.5)), (theta, 0, 2*pi))
print(double_integral)
</execute>

Observation:
Out[2]: 245700.5*pi

Assistant:
<thought> Let's round this result to three decimal places using half up rounding rule. </thought>
<execute>
import math
print(round(245700.5*math.pi, 3))
</execute>

Observation:
771890.886

Assistant:
The answer is <solution> 771890.886 </solution>.
