
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 $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$ are three sets of unit vectors in space, and $\overrightarrow{OA}\bot \overrightarrow{OB}$, $\overrightarrow{OA}\bot \overrightarrow{OC}$, the angle between $\overrightarrow{OB}$ and $\overrightarrow{OC}$ is $60^{\circ}$. Point $P$ is an arbitrary point in space, and $|\overrightarrow{OP}|=1$. It satisfies $|\overrightarrow{OP}\cdot \overrightarrow{OC}|\leqslant |\overrightarrow{OP}\cdot \overrightarrow{OB}|\leqslant |\overrightarrow{OP}\cdot \overrightarrow{OA}|$. Find the maximum value of $|\overrightarrow{OP}\cdot \overrightarrow{OC}|$.

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