
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
Consider a city grid with intersections labeled A, B, C, and D. Assume a student walks from intersection A to intersection B every morning, always walking along the designated paths and only heading east or south. The student passes through intersections C and D along the way. The intersections are placed such that A to C involves 3 eastward moves and 2 southward moves, and C to D involves 2 eastward moves and 1 southward move, and finally from D to B requires 1 eastward move and 2 southward moves. Each morning, at each intersection where he has a choice, he randomly chooses whether to go east or south with probability $\frac{1}{2}$. Determine the probability that the student walks through C, and then D on any given morning.
A) $\frac{15}{77}$  
B) $\frac{10}{462}$  
C) $\frac{120}{462}$  
D) $\frac{3}{10}$  
E) $\frac{64}{462}$

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