Group these ideas for the problem below into $low to $high directions. A direction is a distinct way of attacking the problem; ideas in the same direction share their core mechanism. Give each direction a short name and a one-sentence essence. Put every idea in exactly one direction, using its id.

$hint

$frame

Ideas:
$ideas

Return only JSON, with no other text:
{"directions": [{"name": "...", "essence": "...", "ideas": ["i-0001", "i-0004"]}]}
