Can we use bash's conditional operator with assignment operators like this?

Bash reference manual explains the arithmetic operators as follows.

• conditional operator expr ? expr : expr
• assignment = *= /= %= += -= <<= >>= &= ^= |=

Fisrt, this code seems to work well.

a=1; ((a? b=1 : 2 )) #seems to work

But when I use assignment operator after :, I got the attempted assignment to non-variable error.

a=1; ((a? b=1 : c=1)) #attempted assignment to non-variable error

Why we can use only assignment operators before colon?