
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
A mouse is on the below grid:
        \begin{center}
        \begin{asy}
        unitsize(2cm);
        filldraw(circle((0,0),0.07), black);
        filldraw(circle((0,1),0.07), black);
        filldraw(circle((1,0),0.07), black);
        filldraw(circle((0.5,0.5),0.07), black);
        filldraw(circle((1,1),0.07), black);
        draw((0,0)--(1,0));
        draw((0,0)--(0,1));
        draw((1,0)--(1,1));
        draw((0,1)--(1,1));
        draw((0,1)--(0.5,0.5));
        draw((1,0)--(0.5,0.5));
        draw((1,1)--(0.5,0.5));
        draw((0,0)--(0.5,0.5));
        \end{asy}
        \end{center}
         The paths connecting each node are the possible paths the mouse can take to walk from a node to another node. Call a ``turn" the action of a walk from one node to another. Given the mouse starts off on an arbitrary node, what is the expected number of turns it takes for the mouse to return to its original node?

*Lightning 4.2*

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