
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 the moving points $P$, $M$, and $N$ are respectively on the $x$-axis, the circle $\Gamma_{1}$: $(x-1)^{2}+(y-2)^{2}=1$, and the circle $\Gamma_{2}$: $(x-3)^{2}+(y-4)^{2}=3$, find the minimum value of $|PM| + |PN|$.

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