
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 left and right foci of the ellipse $\frac{x^2}{100} + \frac{y^2}{b^2} = 1 (0 < b < 10)$, and $P$ is a point on the ellipse.

(1) Find the maximum value of $|PF_1| \cdot |PF_2|$.

(2) If $\angle F_1PF_2 = 60^{\circ}$ and the area of $\triangle F_1PF_2$ is $\frac{64\sqrt{3}}{3}$, find the value of $b$.

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