7. (10 points) The greatest common divisor of the natural number $b$ and 175 is denoted as $d$. If $176 \times(b-11 \times d+1)=5 \times d+1$, then $b=$ $\qquad$

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【Analysis】Since the greatest common divisor of $b$ and 175 is denoted as $d$, $d$ and 175 are in a multiple relationship. Factorize 175, and then based on $5 d+1=176, 176$ being 1 more than 175, the value of $d$ can be found. Substituting the value of $d$ into $176 \times(b-11 \times d+1)=5 \times d+1$, the value of $b$ can be determined.

【Solution】Solution: Since the greatest common divisor of $b$ and 175 is denoted as $d$, $d$ must be a divisor of 175, and $175=5 \times 5 \times 7$, so $d$ can only be $1,5,7,25,35,175$; Additionally, from $176 \times(b-11 \times d+1)=5 \times d+1$, it is known that $b-11 \times d+1$ is a non-zero natural number, i.e., $b-11 d+1 \geqslant 1$, thus $5 d+1 \geqslant 176 \Rightarrow d \geqslant 35$. Therefore, $d=35$ or 175.

Substituting $d=35$ into $176 \times(b-11 \times d+1)=5 \times d+1$, we get $b=385$.
Substituting $d=175$ into $176 \times(b-11 \times d+1)=5 \times d+1$, we get $176 \times(b-11 \times 175+1)=5 \times 175+1$ $=876$,

which simplifies to $44 \times(b-11 \times 175+1)=219$, the left side is even, and the right side is odd, a contradiction!
Therefore, $d=175$ does not meet the requirements.
Thus, $b=385$.
The answer is: 385.