
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
Given the function $f(x)=2\cos x \cdot \cos(x+\varphi)$ where $|\varphi| < \frac{\pi}{2}$, choose one of the three conditions, Condition 1, Condition 2, or Condition 3, as known to ensure the existence of the function $f(x)$.

$(Ⅰ)$ Find the value of $\varphi$.

$(Ⅱ)$ Find the maximum and minimum values of $f(x)$ on the interval $[-\frac{\pi}{2}, 0]$.

Conditions:

Condition 1: $f(\frac{\pi}{3})=1$

Condition 2: The function $f(x)$ is increasing on the interval $[0, \frac{\pi}{4}]$

Condition 3: For all $x \in \mathbb{R}$, $f(x) \geq f(\frac{2\pi}{3})$

Note: If the chosen condition does not meet the requirements, 0 points will be awarded. If multiple suitable conditions are chosen and answered separately, the score will be based on the first answer.

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