Implement multi-part cost calculator

You are given a price catalog and a list of orders. Implement the following in any language (Python examples acceptable):

Inputs

• price_catalog: mapping from sku (string) -> unit_price (float).
• orders: list of orders; each order has:
  • order_id (string)
  • items: list of { sku: string, qty: integer }

Part 1 — Basic totals

• For each order, compute total_cost = sum(qty * unit_price) over its valid items.
• Validations and corner cases:
  • If an item's sku is not in price_catalog, skip that item and record it in a per-order "missing_skus" list.
  • If qty is non-numeric, non-integer, <= 0, or overflows type limits, treat it as invalid; skip and record in a per-order "invalid_items" list.
  • Do not crash on malformed input; return both results and an "errors" structure.
• Output: list of { order_id, total_cost } sorted by order_id ascending. In Python, show the lambda used for sorting; in other languages, provide the equivalent.

Part 2 — Discounts and tax

• Extend inputs to support optional per-item discounts: discounts: mapping sku -> discount_rate (0.0–1. 0), and optional per-order tax_rate (0.0–1. 0).
• Compute total_cost = (sum over items of qty * unit_price * (1 - discount_rate))) * (1 + tax_rate).
• Validate numeric ranges for discount_rate and tax_rate; out-of-range values should be treated as invalid and reported.
• Allow the caller to choose sorting: by "order_id" or by "total_cost", ascending or descending (demonstrate the sort key/compare function; in Python show the lambda).

Part 3 — Multi-currency and rounding

• Extend orders with optional currency code (e.g., "USD", "EUR"). Given a rates mapping base->target (e.g., usd_per_unit: mapping currency -> rate), convert each order’s total into a target_currency parameter (default "USD").
• Validate that required conversion rates exist and are numeric; report missing or invalid rates.
• Round monetary outputs to two decimals with banker's rounding (or specify the rounding mode you choose and apply it consistently).

Complexity and robustness

• State the time and space complexity of your implementation for each part, and discuss how you would handle very large inputs (streaming, batching, or memory constraints).
• Provide a brief test plan that covers normal cases, missing_skus, invalid qty, extreme numeric values, discount/tax edge cases, sorting correctness, and currency conversion failures.