
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 \\(F_{1}\\) and \\(F_{2}\\) are the two foci of the ellipse \\(C\\): \\(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)\\), and \\(P\\) is a point on the ellipse \\(C\\) such that \\(\overrightarrow{PF_{1}} \perp \overrightarrow{PF_{2}}\\). If the area of \\(\triangle PF_{1}F_{2}\\) is \\(9\\), then \\(b=\\) ______.

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