
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 points be $P = (1, 2), Q = (3, 6), R = (6, 3),$ and $S = (8, 1)$. Construct a quadrilateral by connecting $P$ to $Q$, $Q$ to $R$, $R$ to $S$, and $S$ to $P$. Determine both the perimeter and the area of quadrilateral $PQRS$, where the perimeter can be expressed as $a\sqrt{2} + b\sqrt{5}$ with $a$ and $b$ integers.

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