
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 $a$, $b$, and $c$ are the lengths of the three sides of a certain right-angled triangle, with $c$ being the hypotenuse. If point $(m,n)$ lies on the line $ax+by+2c=0$, then the minimum value of $m^{2}+n^{2}$ is \_\_\_\_\_.

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