Question

A digital payment platform processes a batch of $N_0 = 1{}000$ transactions through four sequential verification stages ($k = 1, 2, 3, 4$). The transition rules for the transaction count $N_k$ at each stage $k$ are defined as follows:

* Stage 1 (Fraud Filter): $10\%$ of the current batch transactions are flagged and removed, after which $50$ system control transactions are added.
* Stage 2 (Currency Verification): $20\%$ of the current batch transactions are each split into two separate micro-transactions (increasing the net transaction count by $20\%$ of the incoming batch), after which $40$ invalid transactions are purged.
* Stage 3 (Settlement Reconciliation): The batch transaction count decreases by $15\%$ due to netting, after which $150$ pending transactions are merged into the batch.
* Stage 4 (Archival Audit): $20\%$ of the current batch transactions are archived (retaining $80\%$ in the active batch), after which $120$ final clearance tokens are added to the batch.

Place the four stages in order of their final batch transaction counts ($N_k$), from lowest transaction count to highest transaction count.