
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 P is a point of intersection between the ellipse $\frac{x^2}{a_1^2} + \frac{y^2}{b_1^2} = 1$ ($a_1 > b_1 > 0$) and the hyperbola $\frac{x^2}{a_2^2} - \frac{y^2}{b_2^2} = 1$ ($a_2 > 0$, $b_2 > 0$), and $F_1$, $F_2$ are the common foci of both the ellipse and hyperbola, if $\angle F_1PF_2 = \frac{\pi}{3}$, find the value of $\frac{b_1}{b_2}$.

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