
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 $A$, $B$, and $C$ that are not collinear, and $O$ is any point outside the plane. If point $P$ determined by $\overrightarrow{OP} = \frac{1}{5} \overrightarrow{OA} + \frac{2}{3} \overrightarrow{OB} + \lambda \overrightarrow{OC}$ lies in the same plane as points $A$, $B$, and $C$, find the value of $\lambda$.

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