See SN 46.3
sati-sambojjhanga remembers Dhamma-teaching, then thinks (vitakka) about that Dharma.
Dhamma-vicaya-sambojjhnaga investigates that Dharma-teaching recollected by sati, by means of vicara (evaluation) paññāya (same word as the sampajano, what B. thanissaro translates as 'alertness').
ātāpi = ...
From Khuddaka Nikaya's The Questions of King Milinda (translation mine):
‘Venerable Nagasena, what is the characteristic mark of vitakka*?’
‘The characteristic mark of vitakka is "placing" (Appanā), Your Majesty.’
‘Give me an example.’
‘For example, Your Majesty, like a carpenter takes well-finished boards of wood and places (appeti) ...