
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 \( f(x) = \max\{x^{2}-1,2\ln x\} \) and \( g(x) = \max\{x+\ln x,ax^{2}+x\} \),
(1) Find the range of the function \( f(x) \) on the interval \( \left[ \frac {1}{2},1 \right] \);
(2) Discuss whether there exists a real number \( a \) such that \( g(x) < \frac{3}{2}x+4a \) holds for all \( x \in (1,+\infty) \). If such an \( a \) exists, find the range of possible values for \( a \); if not, explain why.

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