
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 points $A(2,3), B(5,4), \overrightarrow{AC}=(3\lambda,5\lambda) (\lambda \neq 0)$, and $\overrightarrow{AP}=\overrightarrow{AB}+\overrightarrow{AC}$. If point $P$ lies on the angle bisector in the first and third quadrants, then the value of $\lambda$ is ______.

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