
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 that $f(x)$ is an odd function defined on $[-2,2]$, and when $x \in [-2,0)$, $f(x)=-ax^{2}-\ln (-x)+1$, where $a \in \mathbb{R}$.  
$(1)$ When $a= \frac {1}{2}$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;  
$(2)$ If for any $x$ in $(0,2]$ it holds that $|f(x)+x| \geqslant 1$, find the maximum value of the real number $a$.

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