
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
Let $S(n)$ and $T(n)$ respectively be the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$, and $\frac{S(n)}{T(n)}= \frac{3n+2}{4n+5}$. Let point $A$ be a point outside the line $BC$, and point $P$ be a point on the line $BC$, and $\overrightarrow{AP}= \frac{a_1+4}{b_3} \cdot \overrightarrow{AB}+\lambda \cdot \overrightarrow{AC}$. Determine the value of the real number $\lambda$.

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