$(20$ points$)$ Continue the setup of the previous problem.

$($a$)$ Suppose A uses a pooling strategy of challenge for both types.

What does $B$ believe about $A\text{'}$s type following a challenge?

$($b$)$ Continuing the previous sub$-$part, what is $B\text{'}$s best response to a

challenge?

$($c$)$ Continuing again, and given $B\text{'}$s best response to challenge found

above, is it optimal for both types of $A$ to challenge?

$($d$)$ Is there a PBE in which A challenges for both types?