
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
A function \( f(x) \) is defined on \(\mathbf{R}\) and is an odd function. When \( x \geqslant 0 \), \( f(x) = -x^2 + 2x \). Another function \( y = g(x) \) has domain \([a, b]\) and range \(\left[\frac{1}{b}, \frac{1}{a}\right]\), where \( a \neq b \) and \( a, b \neq 0 \). On \( x \in [a, b] \), \( f(x) = g(x) \). Is there a real number \( m \) such that the set \(\{(x, y) \mid y = g(x), x \in [a, b]\} \cap \left\{(x, y) \mid y = x^2 + m\right\} \) contains exactly two elements?

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