
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 $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$, $P$ is a moving point on the hyperbola $C$, $|F_{1}F_{2}|=10$, $|PF_{1}|-|PF_{2}|=6$, find the distance $\sqrt{d_1d_2}$ where $d_1$ and $d_2$ are the distances from point $P$ to the two asymptotes of the hyperbola $C$.

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