
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
I1.1 Find \( a \) if \( 2t+1 \) is a factor of \( 4t^{2}+12t+a \).

I1.2 \( \sqrt{K} \) denotes the nonnegative square root of \( K \), where \( K \geq 0 \). If \( b \) is the root of the equation \( \sqrt{a-x} = x-3 \), find \( b \).

I1.3 If \( c \) is the greatest value of \( \frac{20}{b + 2 \cos \theta} \), find \( c \).

I1.4 A man drives a car at \( 3c \ \text{km/h} \) for 3 hours and then \( 4c \ \text{km/h} \) for 2 hours. If his average speed for the whole journey is \( d \ \text{km/h} \), find \( d \).

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