Wa `alaykum as-salamu wa rahmatullahi wa barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dearest brother, thanks a lot for your question. May Allah Almighty favors you with the blessings of Ramadan and grants you the merits of observing fasting, qiyam, i`tikaf, etc.
The last third of the blessed month of Ramadan is the last chance a Muslim should seize. In these days occurs the night of qadr, which is virtuous and greatly blessed to the extent that observing worship in it is better than one thousand months. A Muslim is highly encouraged to spare no efforts in these days to gain Allah's mercy, forgiveness and redemption from the Fire.
In his response to the question, Dr. Muhammad M. Abu Laylah, professor of the Islamic Studies & Comparative Religions at Al-Azhar University, states the following:
The last third of Ramadan is the time for the release from Hell-fire, as the Prophet (peace and blessings be upon him) said in the hadith reported by Salman Al-Farisi: "O people, a great and a blessed month is approaching you. In this month there is a night that is better than one thousand months. Allah has made fasting during its daytime obligatory and prayers at night commendable during this month. Whosoever will do a voluntary good deed during this month, it is as if he performed an obligatory good deed at another time and whosoever will perform an obligatory good deed, it is as if he performed seventy obligatory good deeds at another time. This is the month of patience and the reward of patience is Paradise. This is the month of kindness and charity. In this month the believer’s provision is enlarged. Anyone who will give food to a fasting person in this month, it will bring forgiveness for his sins, will save him from the hellfire and he shall have his reward without the person who is fed losing any of his reward… The first third of the month of Ramadan is the time for rahmah (mercy), the second third is for forgiveness, the third is for the release from Hell-fire…"
The duties of the Muslim in this last third of Ramadan is to try his best to perfect his fasting, to avoid any shortcomings or sins that can harm his fasting, to offer more prayers, preferably in jama`ah (congregation), to advance in reading the Qur'an, to make dhikr (remembrance of Allah), to help others to be more committed to religion, to enter i`tikaf if it is possible for him/her, and to remember that the Prophet (peace and blessings be upon him) used to give extra attention to the last ten days of Ramadan. He could hardly sleep, and his family used also to stay in vigilance with him. The Prophet's generosity in this month and particularly in the last ten days was compared to a fast wind.
We should not miss to remind you and all Muslims to try to settle disputes. It is the best occasion to forgive one another and forget about all enmities and grudges. We should come out of Ramadan in a better shape, understanding, and moral behavior. The Prophet (peace and blessings be upon him) made it clear that "if any Muslim comes out of Ramadan without gaining forgiveness and goodness, he/she is a real loser."
Moreover, Sheikh Muhammad Iqbal Nadvi, Director and Imam of Al-Falah Islamic Center, Oakville, Ontario, Canada, and former Professor at King Saud University, Riyadh, Saudi Arabia, adds:
Some of the duties may be outlined as follows:
1. Purification of one’s soul from all sins and evaluating one’s actions to be always in accordance with the teachings of Islam.
2. Exerting much effort to elevate the soul by doing a lot of good deeds.
3. Acting seriously and using every minute to gain protection from the Hell-fire.
4. Developing a sense of determination in one’s soul that he or she will be always fearful and mindful of Allah and to continue with this spirit after Ramadan.
Allah Almighty knows best.