Tide height by rule of twelfths and cosine, with under-keel clearance windows

# Tide height by rule of twelfths and cosine, and under-keel clearance `input.json` (synthetic, not for navigation) gives two consecutive tide events `event1` and `event2` (`time_min` minutes, `height_m`; either rising or falling), `chart_depth_m`, the boat's `draft_m`, required `min_ukc_m`, `need_minutes`, and `queries` (minutes). Write `solve.py` (Python 3 standard library only); `python solve.py input.json answer.json` writes `answer.json` and `tide.svg`. Fresh inputs are used too. ## Rules Let T = event2 time - event1 time and f = (t - event1 time) / T clamped to [0, 1]. Height = h1 + (h2 - h1) * F(f). - **Twelfths**: split [0, 1] into 6 equal parts; cumulative fractions at the part boundaries are 0, 1, 3, 6, 9, 11, 12 twelfths; interpolate linearly inside a part. - **Cosine**: F(f) = (1 - cos(pi * f)) / 2. - Under-keel clearance UKC = chart_depth + height(twelfths) - draft. A query is `safe` if UKC >= min_ukc. `required_height_m` = draft + min_ukc - chart_depth. - Safe window (per method): scan whole minutes t from event1 to event2 inclusive; a run is consecutive minutes with height >= required height. Report the first run whose length end - start (minutes) is >= `need_minutes` as `{"start", "end"}` (first and last safe minute), or null. - `max_method_diff_m`: the largest |twelfths - cosine| over those whole minutes. Use a 1e-9 tolerance on the >= comparisons. Round heights and UKC to 3 decimals. ## Visual output: tide.svg Static, self-contained SVG (root `<svg>`, no scripts, images, links or external assets; renders headless) with `<polyline id="tide-twelfths">`, `<polyline id="tide-cosine">` and `<line id="min-depth-line">` at the required height. The drawing itself is not graded. ## answer.json `{"required_height_m", "queries": [{"t", "twelfths", "cosine", "ukc", "safe"}, ...], "window_twelfths", "window_cosine", "max_method_diff_m"}` ## Scoring Hidden checks (partial credit; all must pass): `query_heights`, `safe_windows` (exact), `method_difference`, `tide_svg_static` (tide.svg exists, follows the static rules, has the ids/tags above), `fresh_run_1`, `fresh_run_2`. Numbers within 0.0015. 60-second limit per run.