
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
If \( f(t) = 2 - \frac{t}{3} \), and \( f(a) = -4 \), find \( a \).

If \( a + 9 = 12Q + r \), where \( Q, r \) are integers and \( 0 < r < 12 \), find \( r \).

\( x, y \) are real numbers. If \( x + y = r \) and \( M \) is the maximum value of \( xy \), find \( M \).

If \( w \) is a real number and \( 2^{2w} - 2^w - \frac{8}{9}M = 0 \), find \( w \).

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