【Analysis】According to the problem, we can set $\frac{M}{37}=9+\frac{m}{37}(m \leqslant 36)$, and $\frac{1}{37}=0.027, \quad 0.027 \times m \leqslant 0.027 \times$ $36=0.972$, so $\frac{M}{37}$ is a pure repeating decimal with a cycle of 3. The result obtained by Xiaolin is 9.684469, which is the result after rounding to the nearest decimal place, so it consists of 2 cycles, with the same three digits. This cycle should be composed of $6, 4, 8$. Since the digits of the decimal are correct, the last digit 9 is obtained by rounding up from the seventh digit, so the first digit of the cycle should be a number greater than or equal to 5, and the third digit is 8, so this cycle is 648. According to this, solve the problem.

【Solution】According to the problem, we can set $\frac{\mathbb{M}}{37}=9+\frac{\mathrm{m}}{37}(m \leqslant 36)$,
$$
\frac{1}{37}=0.027
$$
$0.027 \times m \leqslant 0.027 \times 36=0.972$,
so $\frac{M}{37}$ is a pure repeating decimal with a cycle of 3 digits.
0.684469 is the result of rounding a 7-digit decimal, so it consists of 2 cycles, and its last digit is obtained by adding 1 to the third digit of the cycle, so the first digit of the cycle is 6, and the correct order of the cycle is 0.648, so the correct calculation result is 9.648649.
Therefore, the answer is: 9.648649.
1. (10 points) Xiaolin performs the following calculation: $M \div 37$, where $M$ is a natural number, and the result is rounded to six decimal places. Xiaolin gets the result 9.684469, which has the correct integer part, and the digits in the decimal part are correct but in the wrong order. The correct calculation result is $\qquad$ .