
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 a right triangle $\triangle AOB$ with an area of $1$, where $O$ is the right angle vertex. Let $\vec{a} = \frac{\vec{OA}}{|\vec{OA}|}$ and $\vec{b} = \frac{\vec{OB}}{|\vec{OB}|}$, and $\vec{OP} = \vec{a} + 2\vec{b}$. Determine the maximum value of $\vec{PA} \cdot \vec{PB}$.

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