
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 maze for mice (see the diagram) is a square $5 \times 5$ meters, where mice can only run along the paths. On two intersections, identical pieces of cheese have been placed (marked with an X). At another intersection, there is a mouse (marked with a circle). The mouse detects both pieces of cheese, but it needs to run the same distance to each piece. Therefore, the mouse cannot decide which piece of cheese to choose and sits thoughtfully in place.

a) Mark another intersection where the mouse could sit thoughtfully (the distance along the paths to both pieces of cheese is the same).

b) Find all 6 intersections where such a mouse could sit (including the ones found earlier).

c) Suggest two intersections where the pieces of cheese could be placed such that the number of suitable places for the thoughtful mouse is maximized.

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