
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 that $a$, $b$, and $c$ are the lengths of the sides of right triangle $\triangle ABC$ with $\angle C = 90^{\circ}$, we call a reciprocal function of the form $y=\frac{a+b}{cx}$ a "Pythagorean reciprocal function". If point $P(1, \sqrt{2})$ lies on the graph of the "Pythagorean reciprocal function" and the area of right triangle $\triangle ABC$ is $4$, then the perimeter of $\triangle ABC$ is ______.

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