
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 circle \\(M\\): \\((x-1)^{2}+(y-3)^{2}=1\\), and circle \\(N\\): \\((x-7)^{2}+(y-5)^{2}=4\\), points \\(P\\) and \\(Q\\) are respectively on circles \\(M\\) and \\(N\\), and point \\(A\\) is a point on the \\(x\\)-axis. The minimum value of \\(|AP|+|AQ|\\) is \_\_\_\_\_\_.

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