
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
Using the Schmidt orthogonalization process, construct an orthonormal basis $\{e_{1}, e_{2}, e_{3}\}$ from the given basis $\{g_{1}, g_{2}, g_{3}\}$:
\[ g_{1}=(1, 1, 0), \quad g_{2}=(2, 0, 1), \quad g_{3}=(0, 1, -2) \]

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