
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 points  $ P,\ Q$  on the parabola  $ y \equal{} x^2$ . Let  $ h$  be the  $ y$  coordinate of the midpoint of the segment  $ PQ$ .

(1) Let  $ L$  and  $ m$  be the length and slope of the segment  $ PQ$  respectively. Express  $ h$  in terms of  $ L,\ m$ .

(2) When  $ L$  is fixed, find the minimum value of  $ h$ .

<span style="color:green">Corrected by moderator.</span>

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