One-time pad without zero: proof check

I started learning cryptography and tried to work through this problem: consider one-time pad where $\mathcal{M}=\mathcal{C}=\{0,1\}^n$ and $\mathcal{K}=\{0,1\}^n\setminus 0^n$ (call this scheme $\Pi$). Find $\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1]$.

My attempt: $\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1]$=$\frac{1}{2}\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1\mid b=0] + \frac{1}{2}\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1\mid b=1]$.

Focus on $\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1\mid b=0]$. The "problematic" case is when the ciphertext is $m_1$ because the adversary knows for sure that in this case $b=0$. in all other cases of the ciphertext, this behaves like regular OTP and thus the best the adversary can do is flip a coin. Formally: $$\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1\mid b=0]=\Pr[c=m_1]+\frac{1}{2}\Pr[c\neq m_1]$$ but $\Pr[c=m_1]=\Pr[k=m_1\oplus m_0]=\frac{1}{|\mathcal{K}|}$ so: $$\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1\mid b=0]=\frac{1}{2}+\frac{1}{2|\mathcal{K}|}$$ the same exact argument can be done when $b=1$ so finally: $$\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1]=\frac{1}{2}+\frac{1}{2|\mathcal{K}|}$$

Is it correct?

Edit:

No your reasoning is wrong. If I consider an attacker $\mathcal{A}$ which output $0$ during any execution, we obtain $\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1\mid b=0]=1$. Then your computation $\Pr[\text{PrivK}_{\mathcal{A},\Pi}^{eav}=1]=\frac{1}{2}+\frac{1}{2|\mathcal{K}|}$ is wrong.

Remember all the intermediates equalities should be true for any adversary

Hint: Do not cut relatively to $b$.