
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
In rectangle \(ABCD\), where \(AB = 3\) and \(BC = 1\), point \(P\) moves along side \(CD\). Let \(\angle PAB = \alpha\) and \(\angle PBA = \beta\). Find the maximum value of \(\frac{\overrightarrow{PA} \cdot \overrightarrow{PB}}{\cos (\alpha + \beta)}\).

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