
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
If point $P$ is on the curve $C_1: \frac{x^2}{16} - \frac{y^2}{9} = 1$, point $Q$ is on the curve $C_2: (x-5)^2 + y^2 = 1$, and point $R$ is on the curve $C_3: (x+5)^2 + y^2 = 1$, then the maximum value of $|PQ|-|PR|$ is _._

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