Triangular FX Arbitrage Path Reconstruction

Given a list of $V$ currency codes currencies and a full $V \times V$ matrix rates where rates[i][j] is the exchange rate from currency $i$ to currency $j$, determine if a compounding currency arbitrage cycle exists.

If an arbitrage opportunity exists, reconstruct the exact ordered sequence of currency codes forming the cycle (e.g., ['USD', 'EUR', 'GBP', 'USD']) and compute the realized multiplicative return factor $\prod R_{ij} > 1.0$.

Your Task

Implement reconstruct_arbitrage_cycle(currencies: list[str], rates: list[list[float]]) -> tuple[bool, list[str], float] that returns:

1. has_arbitrage: boolean True if an arbitrage cycle exists, False otherwise.
2. cycle: list of currency ticker strings starting and ending with the same currency (empty list if no arbitrage).
3. profit_factor: float compounding return (rounded to 6 decimal places; returns 1.0 if no arbitrage).

Constraints

• $2 \le V \le 50$
• $rates[i][j] > 0.0$
• $rates[i][i] == 1.0$

Examples

currencies = ['USD', 'EUR', 'GBP']
rates = [
    [1.0, 0.85, 0.75],
    [1.18, 1.0, 0.88],
    [1.34, 1.14, 1.0]
]
# Cycle USD -> EUR -> GBP -> USD: 0.85 * 0.88 * 1.34 = 1.00232 > 1.0
# Output: (True, ['USD', 'EUR', 'GBP', 'USD'], 1.00232)