
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
Let $f(x) = (x+1)\ln(x+1)$ and $g(x) = ax^2 + x$ where $a \in \mathbb{R}$.  
$(1)$ Find the minimum value of $f(x)$.  
$(2)$ If $\forall x \geq 0$, $f(x) \leq g(x)$, find the range of real number $a$.

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