
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 the parabola $y^2 = 2x$, and point $P$ is any point on the parabola. The projection of $P$ on the $y$-axis is $Q$, and point $M$ is $(2, 3)$. The minimum value of the sum of the lengths of $PQ$ and $PM$ is \_\_\_\_\_\_.

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