
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
Drop perpendiculars IP and IQ from point I onto sides BC and AB respectively. Points P and Q are points of tangency of these sides with the inscribed circle, and the quadrilateral PIQB is a square. The angles KIL and PIQ are right angles, hence angles PIK and QIL are equal. Consequently, right triangles PIK and QIL are congruent. According to Thales' theorem, the length KP=QL is half of the length BP=BQ, and the length AQ is twice the length BQ=BP.

Therefore, the length of side AB is \( AL + LB = \frac{6}{5}AL = \frac{6}{5}AO = \frac{3}{5}AC \). From the Pythagorean theorem, \( BC = \frac{4}{5}AC \). Hence, \( AB:BC:CA = 3:4:5 \).

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