<｜begin▁of▁sentence｜>### Instruction:
In the course of examining public policy related to data-driven decision making, a policy researcher is scrutinizing a database that comprises 150 records, each featuring 3 distinct variables. This compendium is theorized to encapsulate three separate subgroups. To recognize the inherent structures within the data and establish the number of clusters that most effectively captures this segmentation, the researcher contemplates employing the KMeansCluster technique from graspologic.

Equipped with the actual category labels for these records, designated as 'y', the researcher possesses a valuable benchmark for assessing the accuracy of the clustering. Naturally, in broader policymaking circumstances, such explicit labels may not be readily available. Therefore, the researcher anticipates evaluating up to five potential cluster subdivisions, aiming to discern the optimum count based on the highest calculated silhouette coefficient or, alternatively, the adjusted Rand index (ARI) when comparison with true labels is feasible.

How might this policy researcher proceed to expertly apply the KMeansCluster from graspologic to this scenario in order to substantiate the optimal clustering scheme, effectively guiding subsequent policy recommendations or reforms?
### Response:
import numpy as np
from graspologic.cluster import KMeansCluster

# Generate synthetic data with 3 features
np.random.seed(42)
X = np.random.rand(150, 3)

# Known labels (if available, otherwise set y to None)
y = np.array([0] * 50 + [1] * 50 + [2] * 50)

# Initialize and fit the KMeansCluster with a maximum of 5 clusters
clusterer = KMeansCluster(max_clusters=5, random_state=42)
clusterer.fit(X, y)

# Retrieve the optimal number of clusters and the corresponding silhouette scores
optimal_clusters = clusterer.n_clusters_
silhouette_scores = clusterer.silhouette_list_

# Predict the clusters using the best model
predicted_labels = clusterer.predict(X)

# Print the results
print('Optimal number of clusters:', optimal_clusters)
print('Silhouette scores:', silhouette_scores)
print('Predicted labels:', predicted_labels)
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