
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 spider has created a web in the forest, attaching long threads to the branches. A butterfly has flown into this web. Each move, the butterfly or the spider can move along a segment of thread to an adjacent intersection point; the butterfly also has the option to reach the end of a thread (a branch) if it was at an adjacent intersection point. The moves alternate, starting with the butterfly. If the butterfly reaches the branches, it escapes (it wins). If the spider reaches the butterfly, it eats it (and it wins). There is also a possible outcome where no one wins and the game continues indefinitely.

a) What is the outcome of the game in the situation depicted in the picture? (The web has four rings and seven radii.)

b) What is the outcome of the game if the web has three rings and seven radii?

c) What is the outcome of the game if the web has four rings and ten radii?

d) Consider the general case: $K \geq 2$ rings and $R \geq 3$ radii.

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