Therefore, the number of distinct sequences is $ \boxed{1260} $.Question: A technology consultant models data flow efficiency using the function $ f(\theta) = \sin^4 \theta - 4\sin^2 \theta \cos^2 \theta + \cos^4 \theta $. Find the minimum value of $ f(\theta) $ over all real $ \theta $.

["Optimize Data Flow Efficiency: Finding the Minimum Value of a Trigonometric Function", "In modern data systems, optimizing signal efficiency often depends on analyzing complex periodic functions. A recent model used by a leading technology consultant to assess data transmission channels analyzes the function:\n$$\nf(\ heta) = \sin^4 \ heta - 4\sin^2 \ heta \cos^2 \ heta + \cos^4 \ heta\n$$\nThis expression, though rooted in trigonometry, mirrors patterns found in signal modulation and flow dynamics—making its minimum value crucial for performance tuning.", "We aim to find the minimum value of $ f(\ heta) $ over all real $ \ heta $. To simplify, we begin by rewriting $ f(\ heta) $ using algebraic identities.", "---", "Step 1: Group and simplify using identities", "Recall the identity:\n$$\n\sin^4 \ heta + \cos^4 \ heta = (\sin^2 \ heta + \cos^2 \ heta)^2 - 2\sin^2 \ heta \cos^2 \ heta = 1 - 2\sin^2 \ heta \cos^2 \ heta\n$$", "Substitute into $ f(\ heta) $:\n$$\nf(\ heta) = (\sin^4 \ heta + \cos^4 \ heta) - 4\sin^2 \ heta \cos^2 \ heta = (1 - 2\sin^2 \ heta \cos^2 \ heta) - 4\sin^2 \ heta \cos^2 \ heta\n$$\n$$\nf(\ heta) = 1 - 6\sin^2 \ heta \cos^2 \ heta\n$$", "---", "Step 2: Use double-angle identity", "Recall:\n$$\n\sin(2\ heta) = 2\sin\ heta\cos\ heta \Rightarrow \sin^2(2\ heta) = 4\sin^2\ heta\cos^2\ heta\n\Rightarrow \sin^2\ heta\cos^2\ heta = \frac{1}{4}\sin^2(2\ heta)\n$$", "Substitute:\n$$\nf(\ heta) = 1 - 6 \cdot \frac{1}{4} \sin^2(2\ heta) = 1 - \frac{3}{2} \sin^2(2\ heta)\n$$", "---", "Step 3: Minimize the function", "We now minimize:\n$$\nf(\ heta) = 1 - \frac{3}{2} \sin^2(2\ heta)\n$$", "Since $ \sin^2(2\ heta) \in [0, 1] $, the expression $ \frac{3}{2} \sin^2(2\ heta) $ ranges from $ 0 $ to $ \frac{3}{2} $. To minimize $ f(\ heta) $, we maximize $ \sin^2(2\ heta) $. The maximum value is $ 1 $, achieved when $ 2\ heta = \frac{\pi}{2} + k\pi $, i.e., $ \ heta = \frac{\pi}{4} + \frac{k\pi}{2} $.", "Thus, the minimum value is:\n$$\nf_{\ ext{min}} = 1 - \frac{3}{2}(1) = 1 - \frac{3}{2} = -\frac{1}{2}\n$$", "---", "Conclusion", "This model reveals that the lowest data flow efficiency (or highest system loss) predicted by the function reaches $ -\frac{1}{2} $, a critical threshold for optimization. By recognizing the symmetry and periodicity in trigonometric expressions, technology consultants can more effectively tune systems and reduce signal degradation.", "$$\n\boxed{-\frac{1}{2}}\n$$"]