
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
A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 eastward and 2 southward moves, from $C$ to $D$ have 2 eastward and 1 southward move, and from $D$ to $B$ have 1 eastward and 2 southward moves. If at each intersection where the student has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south, find the probability that through any given morning, he goes through $C$ and then $D$.
A) $\frac{15}{77}$
B) $\frac{90}{462}$
C) $\frac{10}{21}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$

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