
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 the coordinates of the vertices of triangle $\triangle ABC$ as $A\left(x_{1}, y_{1}\right)$, $B\left(x_{2}, y_{2}\right)$, and $C\left(x_{3}, y_{3}\right)$, with the lengths of the sides being $a$, $b$, and $c$. Find the coordinates of the incenter $I$ of this triangle.

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