
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 two circles $C_1: x^2 + y^2 = 1$ and $C_2: (x - 2)^2 + (y - 4)^2 = 1$, a moving point $P(a,b)$ is used to draw tangents $PM$ and $PN$ to circles $C_1$ and $C_2$ respectively (with $M$ and $N$ as the points of tangency). If $PM = PN$, find the minimum value of $\sqrt{a^2 + b^2} + \sqrt{(a - 5)^2 + (b + 1)^2}$.

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