
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
Given the arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with the sum of their first $n$ terms being $S\_n$ and $T\_n$ respectively, and $\frac{S\_n}{T\_n} = \frac{3n}{2n+9}$. If point $D$ divides the side $BC$ of $\triangle ABC$ into two parts $a\_n$ and $b\_n$ ($a\_n = BD$, $b\_n = CD$), find $\frac{S\_{\triangle ABD}}{S\_{\triangle ACD}}$ when $n = 5$.

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