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.

Expert feedback:
This is GOOD. You have successfully retrieved the birth year of Li Bai from the wiki search result. Next you should try to solve the calculus problem with the information you obtained.

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

Expert feedback:
This is GOOD. You successfully converted the double integral into a form that can be calculated. Now you can convert the answer to meet the output format requirement.

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

Expert feedback:
This is GOOD. You have got the solution! You can try to submit the solution now.

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