
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 function $h(x) = (m^2 - 5m + 1)x^{m+1}$ is a power function and an odd function.  
(1) Find the value of $m$;  
(2) Find the range of the function $g(x) = h(x) + \sqrt{1 - 2h(x)}$ when $x \in \left[0, \frac{1}{2}\right]$.

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