
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
Find all triplets \((x, y, z)\) of real numbers that are solutions to the equation

$$
\sqrt{3^x \left(5^y + 7^z\right)} + \sqrt{5^y \left(7^z + 3^x\right)} + \sqrt{7^z \left(3^x + 5^y\right)} = \sqrt{2} \left(3^x + 5^y + 7^z\right)
$$

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