
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 $f(x)= \begin{cases}x^{2}+ax+1-a & (x\geqslant 0) \\ f(x+2) & (x < 0)\end{cases}$.

(I) If $a=-8$, find the maximum value of $|f(x)|$ when $-6\leqslant x\leqslant 5$.

(II) For any real number $a$ ($-2\leqslant a\leqslant 4$), there exists a maximum positive number $M(a)$ such that $|f(x)|\leqslant 3$ always holds when $x\in[0,M(a)]$. Find the maximum value of $M(a)$ and its corresponding $a$.

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