
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
Three positive reals  $x , y , z $  satisfy  $x^2 + y^2 = 3^2 
y^2 + yz + z^2 = 4^2 
x^2 + \sqrt{3}xz + z^2 = 5^2 .$  
Find the value of  $2xy + xz + \sqrt{3}yz$ 

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