
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 \( f(x) = ax^2 + 8x + 3 \) where \( a < 0 \), find the positive number \( l(a) \) such that the inequality \( |f(x)| \leq 5 \) holds in the interval \([0, l(a)]\). What value of \( a \) maximizes \( l(a) \), and what is the maximum \( l(a) \)?

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