
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 the function
$$
f(x) = |a \sin x + b \cos x - 1| + |b \sin x - a \cos x| \quad (a, b \in \mathbf{R})
$$
attains a maximum value of 11, then $a^{2} + b^{2} = \, \, \, $ .

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