Finding the posterior distribution of mean and variance given data sample using Gibbs Sampling?

I have the following hierachical bayesian model -
$$\mathbf{x}|\mathbf{c},\sigma^2 \sim \mathcal{N}(\mathbf{x}|\mathbf{c},\sigma^2)$$
$$\mathbf{c}|\mathbf{c}_1,\sigma^2_2 \sim \mathcal{N}(\mathbf{c}|\mathbf{c}_1,\sigma^2_2)$$
$$\sigma^2 \sim \mathcal{U}(\sigma^2|r_1,r_2)$$.

Here, $$\mathbf{x}$$ is a data sample. $$r_1$$, $$r_2$$, $$c_1$$ and $$\sigma_2^2$$ are fixed known values. $$\mathcal{U}(\cdot)$$ and $$\mathcal{N}(\cdot)$$ are uniform and normal distributions respectively. The goal is to generate samples from the posterior distribution $$p(\mathbf{c},\sigma^2|\mathbf{x})$$.