
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  $t$  be a positive constant. Given two points  $A(2t,\ 2t,\ 0),\ B(0,\ 0,\ t)$  in a space with the origin  $O$ . Suppose mobile points  $P$  in such way that  $\overrightarrow{OP}\cdot \overrightarrow{AP}+\overrightarrow{OP}\cdot \overrightarrow{BP}+\overrightarrow{AP}\cdot \overrightarrow{BP}=3.$ Find the value of  $t$  such that the maximum value of  $OP$  is 3.

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