Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \sqrt{2} \)? Find the value of \( x \).

Let's think step by step and output the final answer within \boxed{}.
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