
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 quadratic function \( f(x) = x^2 + x \), and the inequality \( f(-x) + f(x) \leq 2|x| \) which defines a solution set \( C \):

(1) Find the set \( C \).

(2) If the equation \( f(a^x) - a^{x+1} = 5 \) (where \( a > 0 \) and \( a \neq 1 \)) has a solution in \( C \), find the possible values of the real number \( a \).

(3) Let the range of \( f(x) \) over \( C \) be \( A \). If the range of \( g(x) = x^3 - 3tx + \frac{t}{2} \) for \( x \in [0, 1] \) is \( B \), and \( A \subseteq B \), find the possible values of the real number \( t \).

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