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The following is rough and ready so it may containe errors......

I took a sight when the sun was approximately due west using an AH. The purpose was to obtain long (to within a degree or two) for an assumed position.

Checking IE:

29' on

38' off

From which IE = 4.5' on and sun's SD =16.8'

Observation was 0456  26/1/2020  H = 60° 40'

Less IE = 60°  35.5'

H/2  = 30°  18'

Less SD and refraction    30° - 16 - 1 = 30°  01'

From Inmans (1932) the dec is 19°S.

Also from Inmans te EOT = -12 min  =  3°

Approx GHA is 5 X  15  + 180°  - EOT = 75°  + 180° - 3° =252°   (check: 252° from almanac)

cos Z = sin t   cos d  sec h   where Z is the azimuth angle. Because the sun is approximately due west  I will use Z=90.

Hence  sin t =  cos (30) / cos(19)

   and t = 66°

long E =  t   - GHA  = 186°

Hence my assumed long should be in the range  184° (from previous noon method) to 186°. A round number of degrees will make use of 214 or 229 easier.

Lokking at the globe the longitude  line passes through the land mass of the South islad of New Zealand. I will select 43°S as the assumed AP.  Thus from not knowing where in the world I was, apart from being on terra firma, not having an up to date almanac and using approximations  now know that  43S  174°E is a sensible AP.

My next step will be to work a sight by the intercept method. I will then probably cheat and us google earth to measure the AP to GPS distance. If it equals the intercept  will have succeeded.