
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 that $\overrightarrow {a}$ and $\overrightarrow {b}$ are two mutually orthogonal unit vectors in a plane, and another vector $\overrightarrow {c}$ in this plane satisfies $(3 \overrightarrow {a}+ \overrightarrow {c})\cdot (4 \overrightarrow {b}- \overrightarrow {c})=0$, and makes $| \overrightarrow {c}- \overrightarrow {b}|≤k$ hold true. Find the minimum value of $k$.

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