
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
During a chess competition, the final match is held between two players, A and B. The match uses a scoring system where winning a game earns 2 points, drawing earns 1 point, and losing earns 0 points. The match consists of five games, and if a player's score exceeds 5 points, the match concludes; otherwise, it continues. Based on past experience, the probability of A winning each game is $\frac{1}{2}$, while the probability of B winning each game is $\frac{1}{3}$. Each game's outcome is independent of the others. If player A's scores in the nth game are denoted as $a_n = 2$ for a win, $a_n = 1$ for a draw, and $a_n = 0$ for a loss, where $n \in \mathbb{N}^*$ and $1 \le n \le 5$, let $S_n = a_1 + a_2 + \ldots + a_n$.  
(1) Find the probability that $S_3 = 5$.  
(2) Find the probability that $S_5 = 7$.

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