
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  $ x,y,z\in\mathbb{N}_{0}$  such that  $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$ .

*Alternative formulation:* Solve the equation  $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$  in nonnegative integers  $ x$ ,  $ y$ ,  $ z$ .

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