
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
Let  $x$  be a real number between 0 and  $\tfrac{\pi}{2}$  for which the function   $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$  obtains its maximum value,  $M$ . Find the value of  $M + 100\cos^2x$ .

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