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I need help getting the following problem

Let $X_1,..,X_n$ be independent $N(\mu,1)$-distributed random variables. Define $\hat{\theta_n}$ as the point of minimum of $\sum_{i=1}^n(X_i-\theta)^4$.

(1) Show that $\hat{\theta}_n \rightarrow \theta_0$ for some $\theta_0$.
(2) Show that $\sqrt{n}(\theta_n - \theta_0)$ converges to normal distribution.

For (1) I have no idea what $\theta$ is and how to start. For (2) I know I have to use asymptotic normality where

$$\sqrt{n}(\hat{\theta}_n - \mu) \rightarrow N \left(0, \frac{P\psi_{\theta_0}}{(P\psi_{\theta_0})^2}\right) $$ where $P()$ stands for an expectation of sorts, but I'm not sure how that one is evaluated. Any help or hints will be appreciated

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