
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
A game developer has set up an "anti-addiction system" for a new game with the following rules:

1. Within 3 hours (including 3 hours) is considered healthy time. The relationship between the cumulative experience value $E$ (unit: $\exp$) obtained by the player during this time and the playing time $t$ (hours) satisfies the equation: $E = t^{2} + 20t + 16a$;
2. Between 3 and 5 hours (including 5 hours) is considered fatigue time. The player does not gain any experience points during this time (i.e., the cumulative experience value remains the same);
3. Beyond 5 hours is considered unhealthy time. The player starts losing experience points, with the amount of loss being directly proportional to the unhealthy time, with a proportionality factor of 50.

(1) When $a = 1$, write down the function relationship $E = f(t)$ between the cumulative experience value $E$ and the playing time $t$, and calculate the cumulative experience value for playing 6 hours.

(2) The game developer defines the ratio of the cumulative experience value $E$ to the playing time $t$ as the "player enjoyment index," denoted as $H(t)$. If $a > 0$, and the game developer wants the "player enjoyment index" of this game to be no less than 24 during healthy time, determine the range of real numbers for $a$.

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