
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 $f(x) = x^2 + 2x$, the graph of function $g(x)$ is symmetric with the graph of $f(x)$ about the origin.  
(1) Find the explicit expression for function $g(x)$.  
(2) Solve the inequality $g(x) \geq f(x) - |x| - 1$.

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