
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
Given a pool table in the shape of a regular 1990-sided polygon $A_{1} A_{2} \cdots A_{1990}$, a ball is hit from a point $P_{0}$ on $A_{1}A_{2}$, strikes a point $P_{1}$ on $A_{2}A_{3}$, and then successively strikes points $P_{2}, P_{3}, \cdots, P_{1989}, P_{1990}$ on $A_{3}A_{4}, A_{4}A_{5}, \cdots, A_{1990}A_{1}, A_{1}A_{2}$. Let $P_{0}A_{2} = \lambda A_{1}A_{2}$ (where $0 < \lambda < 1$) and $\angle P_{1}P_{0}A_{2} = \theta$.

(1) Determine the range of $\theta$.
(2) If $P_{1990} = P_{0}$, find the value of $\theta$.

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