
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
On a convex polygon-shaped billiard table $A_{1} A_{2} \cdots A_{n}$, there are two billiard balls $M_{1}$ and $M_{2}$. From what direction should ball $M_{1}$ be struck so that it successively hits sides $A_{1} A_{2}, A_{2} A_{3}, \cdots, A_{n} A_{1}$ and then hits ball $M_{2}$? (The reflection rule states that the angle of incidence equals the angle of reflection.)

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