<｜begin▁of▁sentence｜>### Instruction:
In the context of a medical research project, we've synthesized a dataset comprising 300 individual data points, which have been algorithmically grouped into 4 distinct clusters with a given level of variance to simulate patient subpopulations for a hypothetical study. We aim to refine our cluster analysis and enhance our understanding of patient stratification by employing an unsupervised machine learning approach. Specifically, we intend to utilize the AutoGMMCluster API from the NetworkX toolkit to elucidate the optimal number of Gaussian mixture components. This determination will be informed by the Bayesian Information Criterion (BIC), a statistical measure that balances the complexity of the model against the goodness of fit. Can you delineate the process by which AutoGMMCluster might be applied in this scenario to automatically ascertain the most suitable number of clusters, as indicated by the lowest BIC score, and then communicate the optimal cluster quantity deduced from the analysis?
### Response:
import numpy as np
from sklearn.datasets import make_blobs
from graspologic.cluster import AutoGMMCluster

# Generating synthetic data
X, y = make_blobs(n_samples=300, centers=4, cluster_std=0.60, random_state=0)

# Using AutoGMMCluster to find the best Gaussian Mixture Model
auto_gmm = AutoGMMCluster(min_components=2, max_components=5, affinity='all', linkage='all', covariance_type='all',
                           random_state=42, selection_criteria='bic')

# Fitting the model to the data
auto_gmm.fit(X)

# Predicting the clusters
labels = auto_gmm.predict(X)

# Number of components with the best BIC
best_components = auto_gmm.n_components_
print(f'Best number of clusters: {best_components}')
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