
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
Investigate the existence of a local extremum for the function \( f\left(x_{1}, x_{2}\right) = x_{2}^{2} - x_{1}^{2} \) under the condition \( x_{1} - 2x_{2} + 3 = 0 \).

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