
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 vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$, where at least one of $\vec{a}$ or $\vec{b}$ is not a zero vector, and a function of real number $t$ defined as $f(t) = |t \vec{a} + \vec{c}|^2 + |t \vec{b} + \vec{d}|^2$, find the value of $t$ that minimizes $f(t)$.

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