
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 \( a < 0 \). For a given negative \( a \), there exists a largest positive real number \( L(a) \) such that the inequality \( |f(x)| \leq 5 \) holds for the entire interval \([0, L(a)]\). Find the value of \( a \) that maximizes \( L(a) \) and determine this maximum \( L(a) \).

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