
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
In the diagram, six circles of radius 1 with centers $P$, $Q$, $R$, $S$, $T$, and $U$ are tangent to one another and to the sides of $\triangle ABC$, as shown. The centers $P$, $Q$, $R$, $S$, $T$, and $U$ form a hexagonal pattern. Each circle is tangent to at least one side of the triangle, with each circle on the boundary being externally tangent to two other circles.
Find the perimeter of triangle $ABC$.

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