
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  $M$  be a set of points in the Cartesian plane, and let  $\left(S\right)$  be a set of segments (whose endpoints not necessarily have to belong to  $M$ ) such that one can walk from any point of  $M$  to any other point of  $M$  by travelling along segments which are in  $\left(S\right)$ . Find the smallest total length of the segments of  $\left(S\right)$  in the cases**a.)**  $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$ .**b.)**  $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$ .

In other words, find the Steiner trees of the set  $M$  in the above two cases.

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