
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
Given unit vectors $a$ and $b$ satisfying $a \perp b$, and vectors $m = 2a - \sqrt{t - 1}b$, $n = ta + b$ ($t$ is a positive real number), determine the minimum value of $m \cdot n$.

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