
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
In $\triangle OAB$, it is known that $|\overrightarrow{OB}|= \sqrt {2}$, $|\overrightarrow{AB}|=1$, $\angle AOB=45^{\circ}$. If $\overrightarrow{OP}=λ \overrightarrow{OA}+μ \overrightarrow{OB}$ and $λ+2μ=2$, then the range of values for the projection of $\overrightarrow{OA}$ on $\overrightarrow{OP}$ is _______.

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