
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

Find the maximum of

$$
|(K \varphi, \varphi)|=\left|\int_{0}^{\pi} \int_{0}^{\pi} K(x, t) \varphi(x) \varphi(t) d x d t\right|
$$

subject to

$$
(\varphi, \varphi)=\int_{0}^{\pi} \varphi^{2}(x) d x=1
$$

where

$$
K(x, t)=\cos x \cos 2 t+\cos t \cos 2 x+1.
$$

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