
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 lines \( PA \) and \( PB \) are tangent to a circle \( \odot O \) with radius 1 at points \( A \) and \( B \), respectively, and \(|PO|=2\),

\[
\overrightarrow{PM} = 2\lambda \overrightarrow{PA} + (1 - \lambda) \overrightarrow{PB}.
\]

If point \( M \) is inside \( \odot O \) (excluding the boundary), find the range of the real number \(\lambda\).

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