
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 \( O \) be the incenter of \( \triangle ABC \), where \( AB = 3 \), \( AC = 4 \), and \( BC = 5 \). The position vector of point \( P \) is given by
\[ \overrightarrow{O P}=x \overrightarrow{O A}+y \overrightarrow{O B}+z \overrightarrow{O C}, \]
with \( 0 \leqslant x, y, z \leqslant 1 \).

Find the area of the region covered by the trajectory of the moving point \( P \).

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