
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
Point $N$ is a moving point on the circle $(x+5)^{2}+y^{2}=1$. In the right triangle $Rt\triangle ABC$ with point $A(3,0)$ as the right-angle vertex, the other two vertices $B$ and $C$ are on the circle $x^{2}+y^{2}=25$, and the midpoint of $BC$ is $M$. Find the maximum value of $|MN|$.

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