
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  $ABCDEF$  be a regular hexagon of side length  $3$ . Let  $X, Y,$  and  $Z$  be points on segments  $AB, CD,$  and  $EF$  such that  $AX=CY=EZ=1$ . The area of triangle  $XYZ$  can be expressed in the form  $\dfrac{a\sqrt b}{c}$  where  $a,b,c$  are positive integers such that  $b$  is not divisible by the square of any prime and  $\gcd(a,c)=1$ . Find  $100a+10b+c$ .

*Proposed by James Lin*

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