How to attain Rama? (Lakshmana, Bharata, Shatrughna)

Rama is Brahman 

Lakshmana, Bharata, Shatrughna refers to each upaya Svarupam to approach Brahman 

Shruti states: 

Prapyasya Brahmano Rupam - Brahman is Prapya that which is attained.

Praptyupayam Phalam Prapte - 

There are 3 ways in Prapatti to attain Brahman 

1. Seshatvam, 

2. Parathantriyam, 

3. Acharya Abhimanam

taking these we can easily decipher the avatara intention+

What Lakshmana did was Seshatvam - A service to Bhagavan which isn't easy that state is comparable to raw gold hardened yet valuable.

Parathantriyam - To long for Bhagavad Service yet remain as an object due to bhagavad agneya. Bharata remained separate from Bhagavan under Bhagavan's command which is even greater a state than Lakshmana.

Lastly Acharya Abhimanam - Shatrughna was always at service of Bharata as he was his Acharya. Of all states this is the greatest and perhaps the difficult of all states. Thus this actually neatly describes their attachment towards each other yet attaining the same phala.

Source: https://twitter.com/BMidou124/status/1583310722007977984

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To explain further, lakShmaNa served bhagavAn according to his own wishes. rAma asked him to stay in Ayodhya but he refused and came to the forest ro serve him. Thus, a bit of independence in his services.

Bharata accepted what rAma said regardless of what he himself preferred+

In a way, this is also a bit of difference in the thinking of bhakti Yogis and prapannAs.

Both sAttvikAs, but prapannAs exhibit more dependence and so are superior.

Another way to look at it is the difference between prapannAs themselves. 

LakshmaNA's urgency to serve bhagavAn like Azhwars seeking moksha (Arta-prapanna), 

while Bharata's patience is like AchAryAs (dR^ipta-prapannAs).

Another reason is that bhagavAn usually appeared in 4 forms as there are 4 vyUha-s. Even Krishna appeared with balarAma (Sankarshana), Pradyumna his son & Aniruddha his grandson. The latter 2 are aMsha-s of the vyUha-s.