
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
Let vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be given. Consider $\mathbf{p}$ such that \[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\] Among all such vectors $\mathbf{p},$ find constants $s, t,$ and $u$ such that $\mathbf{p}$ is equidistant from $s \mathbf{a} + t \mathbf{b} + u \mathbf{c}$ and $\mathbf{c}$. Determine the values of $s, t,$ and $u$.

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