
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
1) Construct a right triangle $ABC$ with legs $AB = a$, $BC = 2a$, where $a$ is an arbitrary segment. Then the hypotenuse of this triangle, according to the Pythagorean theorem, is $AC = a\sqrt{5}$. 

On the hypotenuse $AC$, mark a segment $CD$ equal to $\sqrt{5}$, and from the point $D$, drop a perpendicular $DH$. By the similarity of triangles $DHC$ and $ABC$, we get $\frac{CH}{BC} = \frac{DC}{AC}$. Hence, $CH = 2$.

2) Assuming $t = 2x + 3$, where $x = \frac{t - 3}{2}$, we get:

$$
\left\{
\begin{array}{l}
f(t - 1) + 2\varphi(2t + 1) = \frac{t - 5}{2} \\
f(x - 1) + \varphi(2x + 1) = 2x
\end{array}
\right.
$$

By substituting the variable $x$ with $t$ in the second equation, we obtain the system:

$$
\left\{
\begin{array}{c}
f(t - 1) + 2\varphi(2t + 1) = \frac{t - 5}{2} \\
f(t - 1) + \varphi(2t + 1) = 2t
\end{array}
\right.
$$

Subtracting the second equation from the first, we have $\varphi(2t + 1) = \frac{t - 5}{2} - 2t = \frac{-3t - 5}{2}$. Hence, $f(t - 1) = 2t + \frac{3t + 5}{2} = \frac{7t + 5}{2}$. Therefore, if $2t + 1 = x$, then $t = \frac{x - 1}{2}$ and thus

$$
\begin{aligned}
& \varphi(x) = \frac{-3x + 3 - 10}{4} = \frac{-3x + 7}{4}, \\
& f(x) = \frac{7x + 12}{2}.
\end{aligned}
$$

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