
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
Let $P$ be a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, and let $M$, $N$ respectively be points on the circles $(x+3)^2 + y^2 = 1$ and $(x-3)^2 + y^2 = 4$. Then, find the range of values for $|PM| + |PN|$.

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