General belief is that Shri Rama, Sita, and Lakshmana lived a good part of their 14 years exile in Forests, by raising a thatched cottage (parNa=leaf, ShAla=home, place) on the banks of River Godavari. There are rival Tourist claims. 1. Nasik- Triyambak, in Maharashtra. 2. Bhadrachalam, in Telangana State. It may be unlikely, because distance between Chitrakoot and Nasik is 1115 km. (as per Google Maps). Distance between Chitrakoot and Bhadrachalam is 1145 km. The distance between Bhadrachalam and Rameshwaram is 1175 km, through NH16. Distance between Nasik and Rameshwaram is 1647 km through NH 48. All these distances indicate that in those days of jungles and dense forests everywhere, with no paved roads, and without motor/rail/air transport it is absolutely difficult to cover the terrains. Especially between Chitrakoot and Nasik, or Chitrakoot and Bhadrachalam. This is because Sita is a Princess with a delicate body. Sometimes, Shri Rama used to lift her and carry her on his shoulders, whenever she was not in a position to walk in the thorn-serpant-wild-animal filed katcha paths. The Godavari near which they stayed may be Gupta Godavari which is in Madhya Pradesh, south-west to Chitrakoot.
Agastya's Hermitage, may be between the Atri's Hermitage and Gupt Godavari. Shri Rama, Sita and Lakshmana went to Panchvati- Godavari, as per the advice of Sage Agastya, during the 13th year of their Exile.
KISHKINDHA KAAND`A - BOOK OF KISHKINDHA
Tasya aasiinam nagasya agree
Drakshyatha Aaditya sankaas`am
Agastyam Rishi sattamam.
Sita was abducted by Ravana. Sugriva was directing his Vanaras(monkeys?)to go to the four sides in search of Sita. This verse is a part of the geography narrated by Sugriva to the explorers.
On that mountain's great top sits the Sun-God like shining great sage Agastya.
In this Sugriiva's geography told to Hanuman, Agastya is said to live on Malaya Mountain, near Kaveri River. But Rama's visit to Agastya Aashrama at 3-12-7 does not say anything about Rama seeing the river Kaveri (Cauveri), or Krishnaveni. Did n't Rama see the rivers Kaveri and Krishnaveni? In Hanuman's search of Sita also, there is no mention of Hanuman and Vanaras crossing the river Krishnaveni, Kaveri, though Sugriva directed them to cross. This we have to check up thoroughly. One thing we can conclude: Before visiting Agastya's hermitage, Rama might not have crossed the rivers Krishna veni and Kaveri, because he was yet to establish his cottage at Panchavati, near Godavari River.
Personal views of ybrao-a-donkey, which are not intended to be imposed on others:
At 3-12-7, Rama has already seen Agastya's hermitage.
At 3-13-13, Agastya sayas that Panchavati is 2 yojanas (approx. 20 kms.) away.
Ito dvi yoojanee taata bahu muula phala udakaha
deesho bahu mrigaha s`riimaan Pancavat`i abhivishrutaha.
At 3-13-18, Agastya mentions that Panchavati is on the banks of the river Godavari and that Sita will enjoy it.
Godaavaryaaha samiipee ca Maithilii tatra ramsyatee.
If Godavari is just 20 kms. away from Agastya's hermitage, it can't be on Malaya Hill or near the river Kaveri. Sugriva's geography must be incorrect. Agastya might have lived in Vindhya Hills and not near Kaveri River, Tamil Nadu.
At Chitrakut, the local people have built a Sita's kitchen (rasooyi), near Gupta Godavari. According to Chitrakuta's temple lore, the Gupta Godavari (in a cave) is the Godavari where Sita, Rama and Lakshmana lived.