Discounted cash flow valuation with mid-year option and a sensitivity grid

# Discounted cash flow valuation with a sensitivity grid `input.json` gives forecast free cash flows and valuation assumptions for a fictional company (synthetic). Write `solve.py` (Python 3 standard library only) so that `python solve.py input.json answer.json` computes the valuation. We re-run it on fresh inputs, so do not hard-code anything. ## Input `fcf` (forecast free cash flow for years 1..N), `wacc`, `terminal_growth`, `mid_year` (bool), `net_debt` (can be negative = net cash), `shares`, `ebitda_last` (year-N EBITDA), `sens_wacc`, `sens_growth` (lists). ## Rules 1. Discount year t (t = 1..N) by (1 + wacc) ** t, or by (1 + wacc) ** (t - 0.5) when `mid_year` is true. `pv_fcf` = sum of discounted cash flows. 2. Terminal value (Gordon growth, at the end of year N): `terminal_value` = fcf_N * (1 + g) / (wacc - g). It is always discounted by (1 + wacc) ** N, even with the mid-year convention: `pv_terminal`. 3. `enterprise_value` = pv_fcf + pv_terminal; `equity_value` = enterprise_value - net_debt; `per_share` = equity_value / shares; `terminal_share` = pv_terminal / enterprise_value; `implied_ev_ebitda` = enterprise_value / ebitda_last. 4. If terminal_growth >= wacc, the whole answer is `{"error": "growth_not_below_wacc"}`. 5. `sensitivity`: a list with one row per value in `sens_wacc` (same order); each row has one entry per value in `sens_growth`: the per-share value with that wacc and growth (other inputs unchanged), rounded to 4 decimals, or `null` when growth >= wacc. 6. Round money values to 2 decimals and `per_share`, `terminal_share`, `implied_ev_ebitda` to 6 decimals when writing. ## Output `answer.json` keys: `pv_fcf`, `terminal_value`, `pv_terminal`, `enterprise_value`, `equity_value`, `per_share`, `terminal_share`, `implied_ev_ebitda`, `sensitivity`. ## Scoring Hidden checks (partial credit = fraction passed; all must pass): `pv_of_cash_flows`, `terminal_value`, `ev_equity_per_share` (money within 0.011, the three ratios within 0.000001), `sensitivity_grid` (within 0.0001, nulls exact) on the given input; `fresh_end_of_year` and `fresh_mid_year` (the whole answer, tolerance 0.011); `growth_not_below_wacc` (rule 4). Each run has a 60-second limit.