
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 $P$ is a point on the circle $x^2+y^2=4$ and is not on the coordinate axes, with $A(2,0)$, $B(0,2)$. The line $PA$ intersects the $y$-axis at point $M$, and the line $PB$ intersects the $x$-axis at point $N$. The minimum value of $|AN|+2|BM|$ is \_\_\_\_\_\_.

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