
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 $ABC$, with $c$ being the hypotenuse and $\angle C = 90^{\circ}$. We define a linear function in the form of $y=\frac{a}{c}x+\frac{b}{c}$ as a "Pythagorean linear function." If the point $P(-1,\frac{\sqrt{3}}{3})$ lies on the graph of the "Pythagorean linear function," and the area of right triangle $ABC$ is $2$, then the value of $c$ is ______.

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