
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Chinese chess is a cultural treasure of the Chinese nation and has a long history. It is rich in interest and has simple and easy-to-understand basic rules. It has a deep mass base in China and is the most widely popularized chess game. In a certain region, a Chinese chess competition is held, starting with group matches, with three players in each group, using a single round-robin format (each pair plays only one game). In each game, the winner earns $3$ points, the loser earns $0$ points, and a draw earns each player $1$ point. Based on the points ranking, players advance to the knockout stage. In case of a tie in points, an additional match is played. It is known that players A, B, and C are in the same group. According to past match data statistics, when players A and B compete, the probability of A winning is $\frac{2}{5}$, the probability of a draw is $\frac{1}{5}$; when players A and C compete, the probability of A winning is $\frac{1}{3}$, the probability of a draw is $\frac{1}{3}$; when players B and C compete, the probability of B winning is $\frac{1}{2}$, the probability of a draw is $\frac{1}{6}$. All games are independent of each other. Considering only the three games in the single round-robin, find:  
$(1)$ The expected points of player A;  
$(2)$ The probability that players A and B have the same points.

Present the answer in LaTex format: \boxed{Your answer}
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