
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
Let $\triangle P_0Q_0R_0$ be a triangle with angles $\angle P_0 = 45^\circ$, $\angle Q_0 = 70^\circ$, and $\angle R_0 = 65^\circ$. For each positive integer $n$, define $P_n$ to be the point on $Q_{n-1}R_{n-1}$ such that $Q_{n-1}P_nR_{n-1}$ is a right angle. Similarly, define $Q_n$ on $R_{n-1}P_{n-1}$ and $R_n$ on $P_{n-1}Q_{n-1}$. Determine the smallest $n$ for which $\triangle P_nQ_nR_n$ is right-angled.

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