
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 that $F\_1$ and $F\_2$ are the two foci of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$, and $P$ is any point on the right branch of the hyperbola distinct from the vertex, and $O$ is the coordinate origin. Among the following propositions:

1. The center of the inscribed circle of $\triangle PF\_1F\_2$ must be on the line $x=3$;
2. The center of the inscribed circle of $\triangle PF\_1F\_2$ must be on the line $x=2$;
3. The center of the inscribed circle of $\triangle PF\_1F\_2$ must be on the line $OP$;
4. The inscribed circle of $\triangle PF\_1F\_2$ must pass through $(3,0)$.

The sequence number of the true proposition is \_\_\_\_\_\_.

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